Showing papers in "Quarterly of Applied Mathematics in 2004"
TL;DR: In this article, it was shown that a single strictly convex entropy is sufficient to characterize a Kruzhkov solution for 1-d scalar conservation laws, using the concept of viscosity solution for the related Hamilton-Jacobi equation.
Abstract: We consider stricly convex, 1-d scalar conservation laws. We show that a single strictly convex entropy is sufficient to characterize a Kruzhkov solution. The proof uses the concept of viscosity solution for the related Hamilton-Jacobi equation.
79 citations
TL;DR: For a one-dimensional system of dissipative balance laws endowed with a convex entropy, this paper proved that a constant equilibrium state is asymptotically L 2-stable under a zero-mass initial disturbance.
Abstract: For a one-dimensional system of dissipative balance laws endowed with a convex entropy, we prove, under natural assumptions, that a constant equilibrium state is asymptotically L2-stable under a zero-mass initial disturbance. The technique is based on the construction of an appropriate Liapunov functional involving the entropy and a so-called compensation term.
73 citations
TL;DR: In this paper, the quantum Euler-Poisson model for semiconductors is considered on spatial bounded domain and the well-posedness of steady state solutions is proven under a proposed subsonic condition with quantum effects involved.
Abstract: The quantum Euler-Poisson model for semiconductors is considered on spatial bounded domain. The equations take the form of Euler-Poisson forced by quantum Bohm potential. In [20], the well-posedness of steady-state solutions is proven under a proposed “subsonic” condition with quantum effects involved. In the present paper the local well-posedness of time-dependent solutions is proven for general pressure-density. Under the same “subsonic” condition proposed in [20], the local solutions are proven to exist globally in time and tend to the corresponding steady-state solution exponentially as time grows up.
72 citations
TL;DR: In this paper, the stability of a discrete viscoelastic system with negative stiffness elements both in the time and frequency domains was analyzed by tuning both the amount of negative stiffness in a standard linear solid and driving frequency.
Abstract: We analytically investigate the stability of a discrete viscoelastic system with negative stiffness elements both in the time and frequency domains. Parametric analysis was performed by tuning both the amount of negative stiffness in a standard linear solid and driving frequency. Stability conditions were derived from the analytical solutions of the differential governing equations and the Lyapunov stability theorem. High frequency response of the system is studied. Stability of singularities in the dissipation tan 6 is discussed. It was found that stable singular tan δ is achievable. The system with extreme high stiffness analyzed here was metastable. We established an explicit link for the divergent rates of the metastable system between the solutions of differential governing equations in the time domain and the Lyapunov theorem.
27 citations
TL;DR: In this paper, a mathematical model for analysis of Love waves propagating in stratified anisotropic (monoclinic) media is presented; this model is based on a newly developed Modified Transfer Matrix (MTM) method.
Abstract: A mathematical model for analysis of Love waves propagating in stratified anisotropic (monoclinic) media is presented; this model is based on a newly developed Modified Transfer Matrix (MTM) method. Closed form dispersed relations are obtained for media consisting of one or two orthotropic layers lying on orthotropic substrate. Conditions for existence of Love waves are analyzed. Horizontally polarized shear surface waves of non-Love type are constructed. A numerical algorithm is worked out for obtaining dispersion relations for Love waves propagating in stratified media containing a large number of layers.
24 citations
TL;DR: In this article, a transmission problem for thermoelastic plates was studied and it was shown that the problem is well posed in the sense that there exists only one solution which is as regular as the initial data and that the local thermal effect is strong enough to produce uniform rate of decay of the solution.
Abstract: In this paper we study a transmission problem for thermoelastic plates. We prove that the problem in well posed in the sense that there exists only one solution which is as regular as the initial data. Moreover we prove that the local thermal effect is strong enough to produce uniform rate of decay of the solution. More precisely, there exist positive constants C and γ such that the total energy E(t) satisfies E(t) ≤ CE(0)e−γt.
24 citations
TL;DR: In this article, a generic relaxation for stress constrained optimal design problems is presented by introducing the stress constrained G closure, which is characterized by all G limits together with their derivatives, and a local representation of the set of G limits and their derivatives is developed.
Abstract: A generic relaxation for stress constrained optimal design problems is presented. It is accomplished by introducing the stress constrained G closure. For a finite number of stress constraints, an explicit characterization of the stress constrained G closure is given. It is shown that the stress constrained G closure is characterized by all G limits together with their derivatives. A local representation of the set of all G limits and their derivatives is developed.
22 citations
21 citations
TL;DR: In this paper, an expansion of the product of (c, £) and the prolate spheroidal angular function of the first kind of radial functions of the second kind was obtained from a series of products of the corresponding spherical functions.
Abstract: Alternative expressions for calculating the prolate spheroidal radial functions of the second kind (c. £) and their first derivatives with respect to £ are shown to provide accurate values over wide parameter ranges where the traditional expressions fail to do so. The first alternative expression is obtained from the expansion of the product of (c, £) and the prolate spheroidal angular function of the first kind (c, rj) in a series of products of the corresponding spherical functions. A similar expression for the radial functions of the first kind was shown previously to provide accurate values for the prolate spheroidal radial functions of the first kind and their first derivatives over all (o) parameter ranges. The second alternative expression for Rml (c, £) involves an integral of the product of (c, rf) and a spherical Neumann function kernel. It provides accurate values when £ is near unity and I —mis not too large, even when c becomes large and traditional expressions fail. The improvement in accuracy using the alternative expressions is quantified and discussed.
19 citations
TL;DR: In this article, the authors consider a stochastic control model in which an economic unit has productive capital and also liabilities in the form of debt, and choose investment and consumption controls which maximize total expected discounted HARA utility of consumption.
Abstract: We consider a stochastic control model in which an economic unit has productive capital and also liabilities in the form of debt. The worth of capital changes over time through investment as well as through random Brownian fluctuations in the unit price of capital. Income from production is also subject to random Brownian fluctuations. The goal is to choose investment and consumption controls which maximize total expected discounted HARA utility of consumption. Optimal control policies are found using the method of dynamic programming. In case of logarithmic utility, these policies have explicit forms.
19 citations
TL;DR: In this paper, an initial and boundary values problem derived from a dissipative Fremond model for shape memory alloys is investigated and the existence of a global solution for the abstract version of the evolution problem is proved by use of a semi-implicit time discretization scheme combined with an a priori estimates-passage to the limit procedure.
Abstract: The paper investigates an initial and boundary values problem which is derived from a dissipative Fremond model for shape memory alloys. Existence of a global solution for the abstract version of the evolution problem is proved by use of a semi-implicit time discretization scheme combined with an a priori estimates-passage to the limit procedure.
TL;DR: In this paper, the authors explicitly compute the constrained quasiconvexification of the integrand associated with the square of the gradient of the state in a typical optimal design problem in which a volume constraint is enforced.
Abstract: We explicitly compute the constrained quasiconvexification of the integrand associated with the square of the gradient of the state in a typical optimal design problem in which a volume constraint is enforced.
TL;DR: A unified algebraic approach to the stabilization of a variety of parabolic boundary control systems in the special case where the semigroup generator admits a Riesz basis, based on the so-called identity compensator is proposed.
Abstract: We study the stabilization problem of linear parabolic boundary control systems. While the control system is described by a pair of standard linear differential operators (, τ), the corresponding semigroup generator generally admits no Riesz basis of eigenvectors. Very little information on the fractional powers of this generator is needed. In this sense our approach has enough generality as a prototype to be used for other types of parabolic systems. We propose in this paper a unified algebraic approach to the stabilization of a variety of parabolic boundary control systems. In the special case where the semigroup generator admits a Riesz basis. we also propose a new and simpler algebraic approach to the stabilization which is based on the so-called identity compensator. To show the usefulness of our approach, a class of linear boundary control systems of second order in t is introduced, to discuss the stabilization or the enhancement of stability of these systems.
TL;DR: In this paper, the authors develop an algorithm to simulate curve straightening flows under which curves in Rra of fixed length and prescribed boundary conditions to first order evolve to elasticae, i.e., to (stable) critical points of the elastic energy given by the integral of the square of the curvature function.
Abstract: Motivated by interpolation problems arising in image analysis, computer vision, shape reconstruction, and signal processing, we develop an algorithm to simulate curve straightening flows under which curves in Rra of fixed length and prescribed boundary conditions to first order evolve to elasticae, i.e., to (stable) critical points of the elastic energy E given by the integral of the square of the curvature function. We also consider variations in which the length L is allowed to vary and the flows seek to minimize the scale-invariant elastic energy Einv, or the free elastic energy E\\. Einv is given by the product of L and the elastic energy E, and Ex is the energy functional obtained by adding a term A-proportional to the length of the curve to E. Details of the implementations, experimental results, and applications to edge completion problems are also discussed.
TL;DR: In this article, a degenerate semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source situated at b was studied, and it was shown that the problem has a unique a* such that a unique global solution u exists for a < a*, and max{u(x,t) : 0 < x < 1} reaches c~ in a finite time for a > a*; this a* is the same as that for q = 0.
Abstract: Let q, a, T, and b be any real numbers such that q > 0, a > 0, T > 0, and 0 < b < 1. This article studies the following degenerate semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source situated at b: xqut uxx = a2S(x b)f(u(x, t)) in (0,1) x (0, T], u(:x, 0) = 0 on [0,1], u(0, t) = u(l, t) = 0 for 0 < t < T, where <5 (x) is the Dirac delta function, / is a given function such that limu^cf(u) = oo for some positive constant c, and f{u) and f'(u) are positive for 0 < u < c. It is shown that the problem has a unique continuous solution u before m&x{u(x,t) : 0 < x < 1} reaches c~, u is a strictly increasing function of t for 0 < x < 1, and if max{u(x,i) : 0 < x < 1} reaches c~, then u attains the value c only at the point b. The problem is shown to have a unique a* such that a unique global solution u exists for a < a*, and max{u(x,t) : 0 < x < 1} reaches c~ in a finite time for a > a*; this a* is the same as that for q = 0. A formula for computing a* is given, and no quenching in infinite time is deduced.
TL;DR: In this article, the authors consider a quasistatic problem of frictional contact between a deformable body and a moving foundation, where the material is assumed to have nonlinear viscoelastic behavior and the contact is modeled with normal compliance and the associated law of dry friction.
Abstract: We consider a quasistatic problem of frictional contact between a deformable body and a moving foundation. The material is assumed to have nonlinear viscoelastic behavior. The contact is modeled with normal compliance and the associated law of dry friction. The wear takes place on a part of the contact surface and its rate is described by the Archard differential condition. The main novelty in the model is the diffusion of the wear particles over the potential contact surface. Such phenomena arise in orthopaedic biomechanics where the wear debris diffuse and influence the properties of joint prosthesis and implants. We derive a weak formulation of the model which is given by a coupled system with an evolutionary variational inequality and a nonlinear evolutionary variational equation. We prove that, under a smallness assumption on some of the data, there exists a unique weak solution for the model.
TL;DR: In this paper, a nonhomogeneous system of pressureless flow was investigated and it was shown that the weak solution is unique under an Oleinik-type entropy condition when the initial data is of measurable function and the solution u(x, t) converges to x as t tends to infinity.
Abstract: In this paper, a nonhomogeneous system of pressureless flow Pt + {pu)x = 0, (pu)t + (pu2)x = px is investigated. It is found that there exists a generalized variational principle from which the weak solution is explicitly constructed by using the initial data; i.e., d2 <92 p(x,t) = --r-^minF(y;x,t), p(x,t)u{x,t) = minF(y;x,t) ox y oxot y hold in the sense of distributions, where F(y; x, t) is a functional depending on the initial data. The weak solution is unique under an Oleinik-type entropy condition when the initial data is of measurable function. It is further shown that the solution u(x, t) converges to x as t tends to infinity. The proofs are based on the generalized variational principle and careful studies on the generalized characteristics introduced by Dafermos [5],
TL;DR: In this article, a path decomposition method is used to obtain pathwise estimates of the approximate solutions, which show the decay properties of the solutions and play a crucial role in proving the stability of the wave-front tracking scheme.
Abstract: We study the model equations of polytropic gas dynamics, which constitute a system of three hyperbolic conservation laws. Global in time BV solutions were obtained by Liu (Indiana Univ Math J 26(1):147–177, 1977) provided that \((\gamma - 1)\) times the total variation of the initial data is sufficiently small; here \(\gamma \) is the adiabatic coefficient. The aim of this paper is to give an alternative proof by exploiting the Dafermos–Bressan–Risebro wave-front tracking scheme. An original feature is the use of the path decomposition method to obtain pathwise estimates of the approximate solutions; these estimates show the decay properties of the solutions and play a crucial role in proving the stability of the wave-front tracking scheme.
TL;DR: In this article, the first Appell's hypergeometric function F1(a, b, c, d;x, y) is considered for large values of its variables x and/or y.
Abstract: The first Appell’s hypergeometric function F1(a, b, c, d;x, y) is considered for large values of its variables x and/or y. An integral representation of F1(a, b, c, d;x, y) is obtained in the form of a generalized Stieltjes transform. Distributional approach is applied to this integral to derive four asymptotic expansions of this function in increasing powers of 1/(1−x) and/or 1/(1−y). For certain values of the parameters a, b, c and d, two of these expansions involve also logarithmic terms in the asymptotic variables 1−x and/or 1−y. Coefficients of these expansions are given in terms of the Gauss hypergeometric function 2F1(α, β, γ;x) and its derivative with respect to the parameter α. All the expansions are accompanied by error bounds for the remainder at any order of the approximation. These error bounds are obtained from the error test and, as numerical experiments show, they are considerably accurate. 2000 Mathematics Subject Classification: 41A60 33C65
TL;DR: A phase transition is a jump discontinuity in a solution u to (1.1) between states u(t, x−) and u( t, x+) belonging to different phases as mentioned in this paper.
Abstract: A phase transition is a jump discontinuity in a solution u to (1.1) between states u(t, x−) and u(t, x+) belonging to different phases. Physical models leading to this setting are provided by liquid vapor phase transitions, elastodynamics or combustion models, see [2, 7, 8, 9, 19, 20] and the references therein. Typically, in the case (1.2) the Riemann problem for (1.1) turns out to be underdetermined and further conditions need to be supplemented. Physically, various criteria have been devised: viscosity [20], viscocapillarity [19] or other kinetic conditions [2]. From an analytical point of view, the above criteria can be described through the generalized kinetic condition
TL;DR: The global existence and asymptotic behavior of smooth solutions to the initial-boundary value problem for the 1-D Lyumkis energy transport model in semiconductor science is studied in this paper.
Abstract: The global existence and asymptotic behavior of smooth solutions to the initial-boundary value problem for the 1-D Lyumkis energy transport model in semiconductor science is studied. When the boundary is insulated, the smooth solution of the problem converges to a stationary solution of the drift diffusion equations, exponentially fast as t —> oo.
TL;DR: In this article, the two dimensional dynamic behavior of a geometrically exact inextensible string is discussed and various exact solutions are derived and various asymptotic theories are derived.
Abstract: The two dimensional dynamic behavior of a geometrically exact inextensible string is discussed. A variety of exact solutions are described and various asymptotic theories are derived. The similarity between the motion of the inextensible string and galactic motion is described.
TL;DR: In this article, the authors characterized the solution space of the boundary value problem as orthogonal complement of a certain set of functions determined by the vector field D in an appropriate Hilbert space.
Abstract: Consider the following boundary value problem in the exterior space 5'd~1 = {x e : |x| > 1} of a sphere in two and three dimensions (d — 2,3): Given a vector field D : Sd~1 —> we ask for all harmonic vector fields B : Sd~l —* IRd which decay at least as fast as a dipole field at infinity and are parallel to D on 5d_1; i.e. there is / : Sd~l —> R such that B = / D. For d = 3, this problem is related to the problem of reconstructing the geomagnetic field outside the earth from directional data measured on the earth's surface. The question for uniqueness or non-uniqueness is of particular interest here. In this paper we characterize the solution space of the boundary value problem as orthogonal complement of a certain set of functions determined by the vector field D in an appropriate Hilbert space. Based on the Hilbert space approach we determine and its dimension dim for certain classes of vector fields D. In particular, we find in d = 2 for those fields D^v which are obtained by restricting a 2jV pole field on S1, dim VpN — 2(N — 1) + 1. This result is robust in the sense that perturbations of Djy which are small in a certain norm do not change the dimension of the solution space. In d = 3 we consider only the axisymmetric situation. Here, we find in the case that D is given by polynomials of order not larger than N the upper bound dim < N and in the 2^—pole case dim = N. For N = 1 (dipole field) the result is proved to be robust, which implies uniqueness of the boundary value problem for all vector fields D close to Di. For Djv with N > 2 it is shown that uniqueness can be enforced if either the Hilbert space is truncated or if stronger decay conditions at infinity are imposed.
TL;DR: In this article, the authors showed that the error of the inviscid solution is bounded by O(e 1 / 2 + e 1 /2 + e ǫ + e) in L 1 -norm.
Abstract: This paper is concerned with global error estimates for viscosity methods to initial-boundary problems for scalar conservation laws u t + f(u) x = 0 on [0, ∞) x [0, ∞), with the initial data u(x, 0) = u o (x) and the boundary data u(0, t) = u_, where u_ is a constant, u 0 (x) is a step function with a discontinuous point, and f E C 2 satisfies f" > 0, f(0) = f'(0) = 0. The structure of global weak entropy solution of the inviscid problem in the sense of Bardos-Leroux-Nedelec [11] is clarified. If the inviscid solution includes the interaction that the central rarefaction wave collides with the boundary x = 0 and the boundary reflects a shock wave, then the error of the viscosity solution to the inviscid solution is bounded by O(e 1 / 2 + e‖lne‖ + e) in L 1 -norm. If the inviscid solution includes no interaction of the central rarefaction wave and the boundary or the interaction that the rarefaction wave collides with the boundary and is absorbed completely or partially by the boundary, then the error bound is O(e‖lne‖ + e). In particular, if there is no central rarefaction wave included in the inviscid solution, the error bound is improved to O(e). The proof is given by a matching method and the traveling wave solutions.
TL;DR: In this article, the Fokker-Planck equation with an added quasilinear term was used for nonlinear filtering with conditional probability density of the state x t given the observation {y(s): 0 < s ≤ t}.
Abstract: In plasma physics, the interaction of radio-frequency waves with a plasma is described by a Fokker-Planck equation with an added quasilinear term. In nonlinear filtering with conditional probability density of the state x t given the observation {y(s): 0 < s ≤ t} is also described by a Fokker-Planck equation with an added first order term. Method for solving Fokker-Planck equation by means of ordinary differential equations is discussed.
TL;DR: In this paper, the authors consider equations of linear and nonlinear thermoelasticity with various boundary conditions and assume radial symmetry of the initial data to prove exponential decay and show the global existence of solutions of the nonlinear problem for small initial data.
Abstract: In this paper we consider equations of linear and nonlinear thermoelasticity with various boundary conditions. We assume radial symmetry of the initial data to prove exponential decay and to show the global existence of solutions of the nonlinear problem for small initial data.
TL;DR: In this article, the authors considered the shock wave in a viscous gas viewed as a (fictitious) strong discontinuity is like undercompressive shock waves in ideal fluids and therefore it is unstable against small perturbations.
Abstract: We discuss the possibility of considering the shock wave in a compressible viscous heat conducting gas as a strong discontinuity on which surface the generalized Rankine-Hugoniot conditions hold. The corresponding linearized stability problem for a planar shock lacks boundary conditions; i.e., the shock wave in a viscous gas viewed as a (fictitious) strong discontinuity is like undercompressive shock waves in ideal fluids and, therefore, it is unstable against small perturbations. We propose such additional jump conditions so that the stability problem becomes well-posed and its trivial solution is asymptotically stable (by Lyapunov). The choice of additional boundary conditions is motivated by a priori information about steady-state solutions of the Navier-Stokes equations which can be calculated, for example, by the stabilization method. The established asymptotic stability of the trivial solution to the modified linearized shock front problem can allow us to justify, at least on the linearized level, the stabilization method that is often used, for example, for steady-state calculations for viscous blunt body flows.
TL;DR: In this article, the large time behavior and the existence of globally defined smooth solutions to the Cauchy problem for the bipolar defocusing nonlinear Schrodinger-Poisson system in the space R 3 were studied.
Abstract: In this paper, we study the large time behavior and the existence of globally defined smooth solutions to the Cauchy problem for the bipolar defocusing nonlinear Schrodinger-Poisson system in the space R 3 .
TL;DR: In this article, the effect of macroscopic nonhomogeneities on the distribution and evolution of strain non-uniformities during the shearing of thermoviscoplastic materials is analyzed.
Abstract: In this paper we present the effect of macroscopic non-homogeneities on the distribution and evolution of strain non-uniformities during the shearing of thermoviscoplastic materials. The thermomechanical parameters (strain hardening, strain rate sensitivity and thermal softening), as well as all the material parameters, are supposed to depend explicitly on the space variable. We show that, even under stability conditions, the strain exhibits intense, time-increasing non-uniformities, following the non-homogeneities, in a specific rate, which depends on the degree of non-homogeneity exhibited by the thermomechanical parameters. By considering both the isothermal and anisothermal cases, we obtain results indicating that non-uniformity measures, based on the control of the strain gradient, are more suitable to give stability conditions of non-homogeneous materials. Moreover, we present numerical results concerning the interplay between material non-homogeneities, initial defects and boundary conditions, for two specific cases: the shearing of a reinforced slab, and the shearing of a
TL;DR: In this paper, it was shown that Lyapunov exponents for concave elemental polygon-polygon-billiards go to zero as the curvature of the concave billiard segments goes to zero.
Abstract: Three results are reported in this work. The first addresses the four 'elemental-polygon' billiards with sides replaced by circular non-overlapping concave elements. Any orbit of the resulting concave billiard is mapped onto a trajectory in the plane that is shown to diverge from the trajectory of the related polygon billiard. This mapping permits application of Lyapunov exponents relevant to an unbounded system to be applied to the bounded concave elemental polygon-billiards. It is shown that Lyapunov exponents for concave elemental polygon-billiards go to zero as the curvature of the concave billiard segments go to zero. The second topic considers the quantum analogue of this problem. A conjecture is introduced which implies that a characteristic quantum number exists below which the adiabatic theorem applies and above which quantum chaos ensues. This parameter grows large as side curvature of the given billiard grows small. Lastly, a correspondence property between classical and quantum chaos for the concave elemental-polygon billiards is described.