Showing papers in "Quarterly of Applied Mathematics in 1999"

Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation

Abstract: We reckon the rate of exponential convergence to equilibrium both in relative entropy and in relative Fisher information, for the solution to the spatially homogeneous Fokker-Planck equation. The result follows by lower bounds of the entropy production which are explicitly computable. Second, we show that the Gross's logarithmic Sobolev inequality is a direct consequence of the lower bound for the entropy production relative to Fisher information. The entropy production arguments are finally applied to reckon the rate of convergence of the solution to the heat equation towards the fundamental one in various norms.

Existence and uniqueness for the linear Koiter model for shells with little regularity

Abstract: We give a simple proof of existence and uniqueness of the solution of the Koiter model for linearly elastic thin shells whose midsurfaces can have charts with discontinuous second derivatives. The proof is based on new expressions for the linearized strain and change of curvature tensors. It also makes use of a new version of the rigid displacement lemma under hypotheses of regularity for the displacement and the midsurface of the shell that are weaker than those required by earlier proofs. Resume. On donne une demonstration simple de l'existence et l'unicite de la solution du modele de Koiter pour des coques minces lineairement elastiques dont les surfaces moyennes peuvent avoir des derivees secondes discontinues. La demonstration est fondee sur de nouvelles expressions des tenseurs linearises de deformation et de changement de courbure. Elle utilise egalement une version nouvelle du lemme du mouvement rigide pour une coque, sous des hypotheses de regularity du deplacement et de la surface moyenne plus faibles que celles des demonstrations anterieures.

The energy transport and the drift diffusion equations as relaxation limits of the hydrodynamic model for semiconductors

Abstract: Two relaxation limits of the hydrodynamic model for semiconductors are investigated. Using the compensated compactness tools we show the convergence of (scaled) entropy solutions of the hydrodynamic model to the solutions of the energy transport and the drift-diffusion equations, according respectively to different time scales.

On a nonlocal dispersive equation modeling particle suspensions

Abstract: We study a nonlocal, scalar conservation law Ut + ((Ka * u)u)x = 0, modeling sedimentation of particles in a dilute fluid suspension, where Ka{x) = a-1 K(x/a) is a symmetric smoothing kernel, and * represents convolution. We show this to be a dispersive regularization of the Hopf equation, Ut + (u2)x = 0, analogous to KdV and certain dispersive difference schemes. Using the smoothing property of convolution and the physical principle of conservation of mass, we establish the global existence of smooth solutions. In physical applications, boundary considerations impose initial data of shock type. Up to shock formation time, we show that solutions remain close to solutions of the Hopf equation. After shock formation time, we give evidence that dispersive, hydrodynamic effects, interacting with Brownian diffusion, can generate oscillatory interface patterns in place of shocks. We discuss possible relevance to the physical phenomenon of layering.

Decay rates of solutions to a von Kármán system for viscoelastic plates with memory

Abstract: We consider the dynamical von Karman equations for viscoelastic plates under the presence of a long-range memory. We find uniform rates of decay (in time) of the energy, provided that suitable assumptions on the relaxation functions are given. Namely, if the relaxation decays exponentially, then the first-order energy also decays exponentially. When the relaxation g satisfies cig1+p(t) < g\\t) <-c0g(t)1+r, 0 < g\"(t) < c2g1+i (t), and g,g1+p g i'(K) with p> 2, then the energy decays as n~+t)F • ^ new Liapunov functional is built for this problem.

Global existence and asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions

Abstract: This paper is concerned with global existence, uniqueness, and asymptotic behavior, as time tends to infinity, of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions. The constitutive assumptions for the Helmholtz free energy include the model for the study of phase transitions in shape memory alloys. To describe phase transitions between different configurations of crystal lattices, we work in a framework in which the strain u belongs to L°°. It is shown that for any initial data of (strain, velocity, absolute temperature) (uo,vo>#o) G L°° x Wq'°° xH1, there is a unique global solution (it, v,6) £ C([0, +oo]; L°°) xC(0, +oo); JTo1'00) fl L°°([0, +oo); W1'00) x C([0, +oo); H1). Results concerning the asymptotic behavior as time goes to infinity are obtained. A new approach is introduced and more delicate estimates are derived to obtain the crucial L°°-norm estimate of u.

On compressible materials capable of sustaining axisymmetric shear deformations. Part 3: helical shear of isotropic hyperelastic materials

Abstract: Conditions on the form of the strain energy function in order that homogeneous, compressible and isotropic hyperelastic materials may sustain controllable static, axisymmetric anti-plane shear, azimuthal shear, and helical shear deformations of a hollow, circular cylinder have been explored in several recent papers. Here we study conditions on the strain energy function for homogeneous and compressible, anisotropic hyperelastic materials necessary and sufficient to sustain controllable, axisymmetric helical shear deformations of the tube. Similar results for separate axisymmetric anti-plane shear deformations and rotational shear deformations are then obtained from the principal theorem for helical shear deformations. The three theorems are illustrated for general compressible transversely isotropic materials for which the isotropy axis coincides with the cylinder axis. Previously known necessary and sufficient conditions on the strain energy for compressible and isotropic hyperelastic materials in order that the three classes of axisymmetric shear deformations may be possible follow by specialization of the anisotropic case. It is shown that the required monotonicity condition for the isotropic case is much simpler and less restrictive. Restrictions necessary and sufficient for anti-plane and rotational shear deformations to be possible in compressible hyperelastic materials having a helical axis of transverse isotropy that winds at a constant angle around the tube axis are derived. Results for the previous case and for a circular axis of transverse isotropy are included as degenerate helices. All of the conditions derived here have essentially algebraic structure and are easy to apply. The general rules are applied in several examples for specific strain energy functions of compressible and homogeneous transversely isotropic materials having straight, circular, and helical axes of material symmetry.

On second sound at the critical temperature

Abstract: Based on low-temperature experimental data in solid dielectric crystals, we derive a model of heat conduction for rigid materials using the theory of thermodynamic internal state variables. The model is intended to admit wavelike propagation of heat below—and diffusive conduction above—a particular temperature value A rapid decay of the speed of thermal waves occurs just below this temperature, coincident with the conductivity of the material reaching a peak. An analysis of weak and strong discontinuity waves is given in order to exhibit several main features of the proposed model.

Existence of solution of a coupled problem arising in the thermoelectrical simulation of electrodes

Abstract: In this paper we prove the existence of a solution to a system of partial differential equations arising from the thermoelectrical modeling of electrodes for electric furnaces. It consists of Maxwell equations coupled with the heat transfer equation through the Joule effect and the fact that thermal conductivity and electrical resistivity depend on temperature. The problem is formulated in cylindrical coordinates to take advantage of its axisymmetry. The result is shown by introducing a regularized problem and using Schauder's fixed point theorem. Passing to the limit requires a priori estimates in weighted Sobolev spaces for an elliptic problem involving a right-hand side that is only integrable.

Rayleigh scattering for the Kelvin-inverted ellipsoid

Abstract: The Kelvin-inverted ellipsoid, with the center of inversion at the center of the ellipsoid, is a nonconvex biquadratic surface that is the image of a triaxial ellipsoid under the Kelvin mapping. It is the most general nonconvex 3-D body for which the Kelvin inversion method can be used to obtain analytic solutions for low-frequency scattering problems. We consider Rayleigh scattering by such a fourth-degree surface and provide all relevant analytical calculations possible within the theory of ellipsoidal harmonics. It is shown that only ellipsoidal harmonics of even degree are needed to express the capacity of the inverted ellipsoid. Special cases of prolate or oblate spheroids and that of the sphere are recovered through appropriate limiting processes. The crucial calculations of the norm integrals, which are expressible in terms of known ellipsoidal harmonics, are outlined in Appendix B.

A new model for acoustic-structure interaction and its exponential stability

Abstract: A new model for the interaction between the acoustic wave in an enclosed air cavity and the transversal motion of a flexible beam is proposed in this paper. This new boundary condition for the coupled wave and Euler-Bernoulli beam equations introduces sufficient damping of the energy of the system to gain uniform exponential stability. Careful physical justification of the boundary condition is based upon well-established theoretical results in acoustics. The estimate of the energy decay rate is obtained using a multiplier technique.

The singular wedge problem in the nonlinear elastostatic plane stress theory

Abstract: A finite elastostatic analysis of the singular equilibrium field in the proximity of the apex of a wedge, with clamped-free radial edges and general far-field loading conditions, is performed. The problem is formulated for compressible hyperelastic sheets under a plane stress condition. An asymptotic procedure is proposed to compute the deformation and stress singular fields. Emphasis is placed on the investigation of the dependence of the order of singularity in the asymptotic Piola-Kirchhoff and Cauchy stresses on the wedge angles. The case of a half-plane bounded to a rigid substrate is studied in detail.

The Atkinson-Wilcox theorem in thermoelasticity

Abstract: An incident disturbance propagates in a thermoelastic medium of the Biot type and it is scattered by a bounded discontinuity of the medium. On the surface of the scatterer any kind of boundary or transmission conditions, that secures well posedness, can hold. The scattered field consists of three kinds of displacement and two kinds of thermal waves. With the exception of one of the displacement waves, namely the transverse elastic wave, all other four scattered waves exhibit exponential attenuation as a result of the coupling between the longitudinal elastic and the thermal disturbances. We show that the displacement field can be expanded in three uniformly and absolutely convergent series in inverse powers of the distance between the observation point and the geometrical center of the scatterer. For the thermal wave a corresponding expansion with two series holds true. Each one of these three elastic and two thermal series describes the corresponding scattered wave and their validity is extended up to the sphere that circumscribes the scatterer. The leading coefficients in the two displacement series of the longitudinal type have only radial components which coincide with the corresponding radial scattering amplitudes. For the transverse displacement series the leading coefficient has only tangential components which coincide with the angular scattering amplitudes. An amazing result, which was not noticed before, is that the thermal scattering amplitudes, appearing as leading coefficients in the thermal expansions, are proportional to the corresponding radial longitudinal amplitudes of the elastic expansions. In other words, both scattering amplitudes of the two thermal waves carry no independent information about the scattering process. Finally, an analytic algorithm is provided which leads to the reconstruction of all five series from the knowledge of the three leading coefficients coming from the expansions for the displacement field alone. Consequently, if the radial and the tangential scattering amplitudes of the displacement field are given in the far field, then the exact displacement and thermal fields can be recovered all the way down to the smallest sphere containing the scatterer. In an equivalent component form we Received January 27, 1998. 1991 Mathematics Subject Classification. Primary 35B40, 35C10, 35K20, 35L20, 73B30, 73D25. ©1999 Brown University 771 772 FIORALBA CAKONI and GEORGE DASSIOS claim that the nine elastic and the two thermal expansions can be completely obtained once the two longitudinal and the two transverse elastic scattering amplitudes are given.

A study of the effective properties of mass and stiffness microstructures—a multiresolution approach

Abstract: The theory of multiresolution decomposition and wavelets is used to study the e ective properties of a thin elastic plate with surface mass density or sti ness heterogeneity, subjected to time-harmonic forcing. The heterogeneity possesses microand macro-scale variations, and has a macroscale outer dimension. It is shown that the microscale mass variation has practically no e ect on the macroscale plate response, whereas microscale sti ness variation can have a signi cant e ect. We derive an e ective constitutive relation pertaining to a microscale sti ness variation. It is shown that it is possible to synthesize classes of di erent sti ness microstructures that have the same footprint on the macroscale component of the plate response. An e ective, smooth, sti ness heterogeneity associated with the classes is developed. The results are rst derived analytically and then supported by numerical simulations.

On the propagation of the bulk of a mass subject to periodic convection and diffusion

Abstract: exist, and u —> 0 and du/dx —> 0 as \\x\\ —+ oo (at rates sufficiently rapid to justify certain integrations by parts). Here u(x, t) is the density, and f(x) is the initial density, of a mass distribution that is convected with velocity a(x) (the drift coefficient) and is subject to diffusion, the positive coefficient b(x) being the diffusion coefficient. Our purpose is to study the manner in which the bulk of the mass is dispersed. The well posedness of the problem is ensured by the standard theory of Co-semigroups, which also guarantees that the positivity of the initial density is preserved [1]. Equation (1) arises in many scientific and technical applications; a recent review of the Fokker-Planck equation has been given by Miyazawa [2], where the relationship of this equation with the Schroedinger equation is discussed and an entire section is

Major simplifications in a current linear model for the motion of a thermoelastic plate

Abstract: A dynamic model for a thin thermoelastic plate proposed by Lagnese and Lions in 1988 [1] has been used recently by several authors (e.g., [2]—[5]) to study existence and stability of solutions to initial/boundary-value problems. Simple, systematic orderof-magnitude arguments show that it is consistent to neglect several terms appearing in the governing differential equations that couple a temperature moment to the average vertical displacement. Further, because the time scale on which the temperature adjusts itself to the strain rate contribution to the energy equation is quite small compared with the longest (isothermal) period of free vibration of the plate, the energy equation can be solved for the temperature in terms of derivatives of the vertical displacement and hence the system reduced to a single equation, only slightly more complicated than the classical (Kirchhoff) equation of motion. Among other things, it is shown that the temperature has a cubic rather than a linear variation through the thickness. Finally, another order-of-magnitude estimate for a clamped aluminum plate of one meter radius and 1mm thickness shows that thermal damping acting alone takes on the order of 200 cycles of vibration to halve the initial amplitude. 1. The equations of linear thermoelasticity. Let xQ, a = 1,2, and z denote Cartesian coordinates in an inertial reference frame and let a homogeneous, isotropic, thermoelastic plate-like body occupy the closure of the open set Q x where fl, the midplane of the plate, is a connected region of the xQ-plane with piecewise smooth boundary dfl and '2H is the plate's thickness. Then, from the standard references by Boley and Weiner [6] or Carlson [7], the linearized governing three-dimensional equations in Cartesian tensor notation comprise the equations of motion, &af3,a Tf3,z f(3 = ; (1*1) Ta,a + + / = pil, (1.2) Received November 10, 1997. 1991 Mathematics Subject Classification. Primary 35B20, 73B30. ©1999 Brown University 673

Conservative energy discretization of Boltzmann collision operator

Abstract: The paper deduces a kinetic model obtained introducing a discretization of the Boltzmann equation based on an equispaced distribution of allowed particle energies. The model obtained is a system of integro-differential equations with integration over suitable angular variables: one over the portion of the unit sphere between two parallels symmetric with respect to the equatorial plane perpendicular to the velocity of the field particle, and one over a unit circle. The model preserves mass, momentum and energy. Furthermore, there exists an ff-functional describing trend toward an equilibrium state described by a Gaussian distribution. Particular attention is paid to the identification of a criterion which indicates the values of the discretization parameters.

Asymptotic analysis of the one-dimensional Ginzburg-Landau equations near self-duality

Abstract: It is known that when the Ginzburg-Landau parameter κ = 1/ √ 2 the one-dimensional Ginzburg-Landau equations exhibit self-duality and may be reduced into a pair of first order ODE. The present asymptotic analysis initially focuses on infinite samples of superconductors for which |κ − 1/ √ 2| œ 1. It is shown that when the value of the applied magnetic field at infinity lies between κ and 1/ √ 2 a superconducting solution exists. It is later shown, for arbitrary values of κ, that no solution, other than the normal state can exist for applied magnetic field values which lie outside the above interval.

Electrodiffusional free boundary problem, in a bipolar membrane (semiconductor diode), at a reverse bias for constant current

Abstract: A singular perturbation problem, modeling one-dimensional time-dependent electrodiffusion of ions (holes and electrons) in a bipolar membrane (semi-conductor diode) at a reverse bias is analyzed for galvanostatic (fixed electric current) conditions. It is shown that, as the perturbation parameter tends to zero, the solution of the perturbed problem tends to the solution of a limiting problem which is, depending on the input data, either a conventional bipolar electrodiffusion problem or a particular electrodiffusional time-dependent free boundary problem. In both cases, the properties of the limiting solution are analyzed, along with those of the respective boundary and transition layer solutions. 0. Introduction. In our recent paper [1] we analyzed the electrodiffusional free boundary problem that arose asymptotically in the singularly perturbed model of electrodialysis for a vanishing perturbation parameter. This model concerned the passage of a specified direct electric current through a layer of univalent electrolyte adjacent to the wall (cathode, cation exchange membrane) selectively permeable to positive ions (cations) only. The simplest version of the governing equations was t > 0: pf = {p£x +pe4>%)x Vx€(0,l), (0.1) t > 0: n\\ = (n% -ne(j)£x)x Vxe(0,l), (0.2) t > 0: e%t+Pex < + (P£ + n£Wx = (0.4) Here I(t) is the electric current density. The first term on the left-hand side of (0.4) stands for the displacement current, whereas the second and the third correspond to the diffusion and conduction current components, respectively. For galvanostatic (fixed current) conditions, considered in Ref. [1], I(t) = I = const (0-5) where I is specified by the boundary conditions. The main result of Ref. [1] consisted in proving that for e —> 0, / > /llm = 4, the solution of the perturbed problem (0.1)-(0.5) with the respective boundary-initial conditions tends to that of the following free boundary problem: ct=cxx, Vx G (0,R(t)), R(t) G (0,1), (0.6) c = 0, Vxe(R{t), 1), (0.7) c(R{t),t) = 0, (0.8) cx(R{t),t) = -1-. (0.9) Here c(x,t)d= limpe(x,t) = lim ne(:r, t). (0.10) £ —>0 e^0 In addition to this result, the limiting problem for I < 7hm has been analyzed, along with the asymptotics for the boundary layers solutions (for both I < Ihm and I > Ihm) and that for the \"empty\" zone R(t) < x < 1, developing for I > 7llm. ELECTRODIFFUSIONAL FREE BOUNDARY PROBLEM 639 The outlined treatment thus addressed one of the few basic functional elements of membrane transport—passage of ions from the electrolyte solution into a charge-selective object (ion-exchange membrane, metal electrode). Another prototypical situation concerns the transfer of ions between different objects of this type, in particular with alternating charge selectivity. This is namely the case in a bipolar membrane—a sandwich formed by an anion exchange membrane (A) adjacent to a cation exchange membrane (B)—the object of our study in this paper. Bipolar membranes are used, in particular, for acid-base generation. Acid-base generation occurs as a result of water electrolysis under the action of strong nonequilibrium electric fields that develop around the A-B junction upon the passage of a specified direct electric current from A to B (see Refs. [3]—[5]). This electric field and the pertaining development of the space charge fronts, irrespectively of the related electrolysis, is the issue that we are concerned with here. The entire setup we are about to study is mathematically identical to that for a reversely biased semiconductor diode operated at a constant current. The simplest relevant time-dependent model problem treated here reads t > 0: p£t = (p£x +P£ 0: n£t = a(n£x n£

Sharp stability estimates for quasi-autonomous evolution equations of hyperbolic type

Abstract: We study the energy decay of the difference of two solutions for dissipative evolution problems of the type: u\" + Lu + g(u') = h(t), t > 0, including wave and plate equations and ordinary differential equations. In the general case, when the damping term g behaves like a power of the velocity v!, the energy decreases like a negative power of time, multiplied by a constant depending on the initial energies. We provide estimates on these constants and prove their optimality. In the special case of the ordinary differential equation with periodic forcing, we establish, relying on a controllability-like technique, that the decay is in fact exponential, even under very weak damping. Resume. On etudie la decroissance de l'energie pour la difference de deux solutions dans des problemes devolution dissipatifs du type: u\" + Lu + g{u') = h{t), t > 0. Ceci s'applique en particulier aux equations des ondes et des plaques et a des equations differentielles ordinaires. Dans le cas general, et lorsque le terme d'amortissement g se comporte comme une puissance de la velocite u', l'energie decroit comme une puissance negative du temps, que multiplie une constante dependant des energies initiales. On donne des estimations sur ces constantes et on prouve leur optimalite. Dans le cas de l'equation differentielle ordinaire avec un terme source periodique, on montre, en utilisant une technique de type controlabilite, que la decroissance en temps est en fait exponentielle, et ce meme en presence d'un amortissement tres faible. Received February 8, 1996. 1991 Mathematics Subject Classification. Primary 35L70, 35L75, 35B35, 35B40; Secondary 34D05. E-mail address: soupletSmath.univ-parisl3.fr ©1999 Brown University 55

A note on the propagation of the bulk of a disturbance for a hyperbolic equation

Abstract: Our arguments can be extended to other linear hyperbolic equations with nonconstant coefficients but, for the sake of brevity, we consider here only the damped wave equation (I)If the initial data correspond to a disturbance that is confined, in the sense that / and g have compact support, then we can ask at what rate the disturbance propagates. The conventional answer is that the disturbance propagates at unit speed, i.e., in such a way that the distance travelled equals the time elapsed. The ground for this assertion is, of course, the entirely correct one that if the supports of / and g are contained within the compact interval a < x < (3 then, for each t > 0, the support of u{x, t) is contained within the interval a — t < x < (3 +1. Our purpose is to point out that the conventional answer nonetheless fails to tell us how the bulk of the disturbance propagates and that, in fact, the bulk propagates in such a way that the distance travelled is proportional to the square root of the time elapsed, as is the case for the parabolic heat equation. (See, for example, Fichera [1], Day [2], [3]. See also the discussion of what happens for the Fokker-Planck equation with periodic coefficients in Day and Saccomandi [4].)

Generalizations of the Greenberg-Rascle construction of periodic solutions to quasilinear equations of 1-D elasticity

Abstract: and, thus, have jump discontinuity in ^ at u = 0. Following the famous papers of Lax [5] and MacCamy and Mizel [6], researchers have collected a large body of evidence supporting the belief that solutions of systems of conservation laws develop shocks.1 This paper deals with shock-free solutions that have a very remarkable property: they are spatially and temporally periodic! The mathematical importance of having a periodic solution is amplified by the following observation. Let (v,u) be a bounded, nonconstant, periodic solution of (WEI) and let

A class of inverse problems for viscoelastic material with dominating Newtonian viscosity

Abstract: Memory kernels in linear stress-strain relations involving a Newtonian viscosity are identified by solving a class of inverse problems. The inverse problems are reduced to nonlinear Volterra integral equations of the first kind which in turn lead to corresponding Volterra equations of the second kind by differentiation. Applying the contraction principle with weighted norms we derive global (in time) existence, uniqueness and stability of the solution to the inverse problems under similar assumptions as for related inverse problems in heat flow.