Showing papers in "Quarterly of Applied Mathematics in 2002"

Stabilization of the Korteweg-De Vries equation with localized damping

Abstract: We study the stabilization of solutions of the Korteweg-de Vries (KdV) equation in a bounded interval under the effect of a localized damping mechanism. Using multiplier techniques we deduce the exponential decay in time of the solutions of the underlying linear equation. A locally uniform stabilization result of the solutions of the nonlinear KdV model is also proved. The proof combines compactness arguments, the smoothing effect of the KdV equation on the line and unique continuation results.

Maximum and minimum free energies for a linear viscoelastic material

Abstract: Certain results about free energies of materials with memory are proved, using the abstract formulation of thermodynamics, both in the general case and as applied within the theory of linear viscoelasticity. In particular, an integral equation for the strain continuation associated with the maximum recoverable work from a given linear viscoelastic state is shown to have a unique solution and is solved directly, using the Wiener-Hopf technique. This leads to an expression for the minimum free energy, previously derived by means of a variational technique in the frequency domain. A new variational method is developed in both the time and frequency domains. In the former case, this approach yields integral equations for both the minimum and maximum free energies associated with a given viscoelastic state. In the latter case, explicit forms of a family of free energies, associated with a given state of a discrete spectrum viscoelastic material, are derived. This includes both maximum and minimum free energies.

On derivative of energy functional for elastic bodies with cracks and unilateral conditions

Abstract: In this paper we consider elasticity equations in a domain having a cut (a crack) with unilateral boundary conditions considered at the crack faces. The boundary conditions provide a mutual nonpenetration between the crack faces, and the problem as a whole is nonlinear. Assuming that a general perturbation of the cut is given, we find the derivative of the energy functional with respect to the perturbation parameter. It is known that a calculation of the material derivative for similar problems has the difficulty of finding boundary conditions at the crack faces. We use a variational property of the solution, thus avoiding a direct calculation of the material derivative.There are many results related to the differentiation of the potential energy functional with respect to variable domains (see, e.g., [9, 4, 5, 16, 18, 17, 3]). The general theory of calculating material and shape derivatives in linear and nonlinear boundary value problems is developed in [6].Derivatives of energy functionals with respect to the crack length in classical linear elasticity can be found by different ways. It is well known that the classical approach to the crack problem is characterized by the equality-type boundary conditions considered at the crack faces [13, 4, 7, 14, 12, 15]. As for the analysis of solution dependence on the shape domain for a wide class of elastic problems, we refer the reader to [8].In the works [1,2] the appropriate technique of finding derivatives of the energy functional with respect to the crack length for unilateral boundary conditions is proposed, which can be used for a wide class of the unilateral problems. Qualitative properties of solutions (solution existence, solution regularity, dependence of solutions on parameters, etc.) in the crack problem for plates, shells, two- and three-dimensional bodies with unilateral conditions on the crack faces are analysed in [2] (see also [20, 19, 10, 11]).

Maximum recoverable work, minimum free energy and state space in linear viscoelasticity

Abstract: The various formulations of the maximum recoverable work used in the literature are proved to be equivalent. Then an explicit formula of the minimum free energy is derived, starting from the formulation of the maximum recoverable work given by Day. The resulting expression is equivalent to that found by Golden and other authors. However, the particular formulation allows us to prove that the domain of definition of minimum free energy is the whole state space. Finally, the maximum recoverable work is shown to be put as the basis of the thermodymamics of viscoelastic materials under isothermal conditions. In this context the usual relation between the Clausius-Duhem inequality and the dissipation of the material is restored.

Theory of exact solutions for the evolution of a fluid annulus in a rotating Hele-Shaw cell

Abstract: Motivated by a series of recent experiments on the evolution of fluid annuli in rotating Hele-Shaw cells, this paper presents a new class of exact time-dependent solutions to a mathematical model of this nonlinear free boundary problem. For a certain class of initial conditions, the free boundary problem is reduced to the solution of a finite set of coupled nonlinear ordinary differential equations. These solutions can be explicitly studied and, despite the fact that the model problem is mathematically ill-posed, display the same qualitative features as the recent experiments. It is discussed how the present exact solutions might form an important basis for further study of the appropriately regularized model problem.

Singular solutions to a nonlinear elliptic boundary value problem originating from corrosion modeling

Abstract: We consider a nonlinear elliptic boundary value problem on a planar domain. The exponential type nonlinearity in the boundary condition is one that frequently appears in the modeling of electrochemical systems. For the case of a disk, we construct a family of exact solutions that exhibit limiting logarithmic singularities at certain points on the boundary. Based on these solutions, we develop two criteria that we believe predict the possible locations of the boundary singularities on quite general domains.

Global solution to a phase field model with irreversible and constrained phase evolution

Abstract: This note deals with a nonlinear system of PDEs describing some irreversible phase change phenomena that accounts for a bounded limit velocity of the phase transition process. The underlying model is thermodynamically consistent as it can be proved by a balance of the internal microscopic forces. An existence result is established by using time discretization, compactness arguments, and techniques of subdifferential operators.

Asymptotic behavior of subsonic entropy solutions of the isentropic Euler-Poisson equations

Abstract: The hydrodynamic model for semiconductors in one dimension is considered. For perturbed Riemann data, global subsonic (weak) entropy solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, is constructed and their asymptotic behavior is analyzed. In subsonic domains, the solution is smooth and, exponentially as t →∞, tends to the corresponding stationary solution due to the influence of Poisson coupling. Along the shock discontinuity, the shock strength and the difference of derivatives of solutions decay exponentially affected by the relaxation mechanism.

Asymptotic and exact fundamental solutions in hereditary media with singular memory kernels

Abstract: A method for constructing time-domain asymptotic solutions of hyperbolic partial differential equations with delay, with singular memory kernels, is presented. The asymptotic solutions are expressed in terms of basis functions that are regularizations of a sequence of distributions related by fractional integration. It is demonstrated that the signal builds up gradually from a zero value after the passage at the wavefront, resulting in signal delay. Attenuation splits into two parts: a frequency-independent amplitude modulating factor and a frequency-dependent part implicit in the basis functions. Explicit basis functions are obtained for the £-1/2 and t~n/3 singularities. Asymptotic solutions are derived for systems of PDEs with the t-1/2 singularity and for scalar equations with the t~n/3 singularities. Asymptotic fundamental solutions are constructed for scalar equations with singular kernels. For two families of scalar equations with singular kernels, asymptotic fundamental solutions are shown to be exact.

Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives

Abstract: Alternative expressions for calculating the prolate spheroidal radial functions of the first kind Rml(1)(c, ζ) and their first derivatives with respect to ζ are shown to provide accurate values, even for low values of l - m where the traditional expressions provide increasingly inaccurate results as the size parameter c increases to large values. These expressions also converge in fewer terms than the traditional ones. They are obtained from the expansion of the product of Rml(1) (c,ζ) and the prolate spheroidal angular function of the first kind Sml(1)(c, η) in a series of products of the corresponding spherical functions. King and Van Buren [12] had used this expansion previously in the derivation of a general addition theorem for spheroidal wave functions. The improvement in accuracy and convergence using the alternative expressions is quantified and discussed. Also, a method is described that avoids computer overflow and underflow problems in calculating Rml(1)(c,ζ) and its first derivative.

Subsonic lamb waves in anisotropic plates

Abstract: A six-dimensional complex formalism for analysis of Lamb waves propagating with subsonic speed in anisotropic plates is formulated. Conditions for nonexistence of certain Lamb waves in anisotropic plates are obtained. An example of a transversely isotropic plate having "forbidden" speed at which no subsonic Lamb wave propagates is presented.

Viscosity solutions for dynamic problems with slip-rate dependent friction

Abstract: The dynamic evolution of an elastic medium undergoing frictional slip is considered. The Coulomb law modeling the contact uses a friction coefficient that is a non-monotone function of the slip-rate. This problem is ill-posed, the solution is nonunique and shocks may be created on the contact interface. In the particular case of the one-dimensional shearing of an elastic slab, the (perfect) delay convention can be used to select a unique solution. Different solutions in acceleration and deceleration processes are obtained. To transform the ill-posed problem into a well-posed one and to justify the choice of the perfect delay criterion, a visco-elastic constitutive law with a small viscosity is used here. An existence and uniqueness result is obtained in three dimensions. The assumptions on the functions implied in the contact model are weak enough to include both the normal compliance and the Tresca model. The following conjecture, based on results of numerical simulations, is stated: in the elastic case, the solution chosen by the perfect delay convention is the one obtained from the solutions of the problem with viscosity, when the viscosity tends to zero.

A mechanism for linear instability in two-dimensional rimming flow

Abstract: In rimming flow, a thin film of viscous liquid coats the inside of a cylinder whose axis is horizontal and which is rotating with constant angular velocity. It has been established experimentally that such flows are often unstable with a variety of secondary flow regimes having been observed experimentally [15]. We use a lubrication approximation extended to the first order in the dimensionless film thickness (including the small effects of the variation of the film pressure across its thickness and the surface tension) and study the stability of the steady solutions to two-dimensional disturbances. The modified evolution equation is found to have both asymptotically stable and unstable solutions arising from the pressure terms. Surface tension effects place a restriction on the critical wave number when instability occurs: in many cases, surface tension prevents instability. 1. The physical problem. The occurrence of instabilities in the flow of a thin film of viscous liquid placed in a cylindrical tube of radius R whose axis is horizontal and which is rotating with constant angular velocity u> (Fig. 1) has attracted considerable interest in the scientific literature [1]—[4]. Such flows are sometimes referred to as rimming flows. The liquid is entrained on the cylinder wall and, given suitable wetting properties, a continuous film can be obtained. Previous authors have investigated this problem both experimentally and theoretically and some progress has been made. In many cases, it is found experimentally that the flow appears to be unstable to both twoand threedimensional disturbances [15]. One well-known example of instability arises when the liquid, instead of settling into a configuration that is uniform in the axial direction, settles into a number of wet and (relatively) dry areas (or rings) along the length of the cylinder. So striking are these features that the term \"hygrocysts\" [1] has been coined to describe this apparently stable secondary state. Visually, the effect vaguely resembles the well-known results obtained by G. I. Taylor for the flow of a viscous liquid in the annular region between two rotating cylinders (see, e.g., [3]). Flows of this type obviously have industrial applications, a typical example being the process of coating the inside of cylindrical fluorescent light bulbs. Such a bulb consists of Received May 22, 2000. 2000 Mathematics Subject Classification. Primary 76D07, 76E17, 76D08, 76B45.

L 2 -regularity theory of linear strongly elliptic dirichlet systems of order 2m with minimal regularity in the coefficients

Abstract: In this article, we consider the following Dirichlet system of order 2m: L(x,∇)u = f(x) in Ω, ∇ku = 0 on δΩ (k = 0,...,m - 1). Here, Ω is a smooth bounded domain in Rn and the differential operator L(x, ∇) given by (1) satisfies the Legendre-Hadamard condition (4). From the general elliptic theory we know that for sufficiently smooth coefficients Aαβ(m), Bαβ(km),Cα(k) and for f ∈ H-m+s(Ω,RN), every weak solution u ∈ H0m(Ω,RN) is actually in Hm+s(Ω,RN) and satisfies an a priori estimate of the following form: ||u||Mm+s(Ω,RN) ≤ Ĉ ||f||H-m+s(Ω,RN) + K||u||L2(Ω,RN). The latter a priori estimate is of particular interest in applications to nonlinear PDEs (see, e.g., [6] and [10]). There the coefficients of L(x, ∇) result from a linearization procedure and consequently they cannot be chosen as smooth as one likes. Therefore, e.g. in [10] (Kato), the author cannot use the famous results stated in [4] (Agmon-Douglis-Nirenberg) but refers to [14] (Milani) instead.

Global attractivity in an RBC survival model of Wazewska and Lasota

Abstract: We obtain a condition for the positive equilibrium to be a global attractor of the survival model of red blood cells proposed by Wazewska and Lasota. Our technique is novel in the sense that a pair of nonlinear equations is utilized, and our result improves earlier results in [3] and [4].

Initial-boundary value problem to systems of conservation laws with relaxation

Abstract: In this paper we consider the initial-boundary value problem (IBVP) for the one-dimensional Jin-Xin relaxation model. The main interest is to study the boundary layer behaviors in the solutions to the IBVP of the relaxation system and their asymptotic convergence to solutions of the corresponding hyperbolic conservation laws in the limit of small relaxation rate. First we develop a general expansion theory for the relaxation IBVP using a matched asymptotic analysis. This formal procedure determines a unique equilibrium limit, and also reveals rich initial and boundary layer structures in the solutions of the relaxation system. Arbitrarily accurate solutions to the IBVP of the relaxation system are then constructed by combining the various orders of the equilibrium solutions, the initial and boundary layer solutions. The validity of the initial and boundary layers and the asymptotic convergence results are rigorously justified through a stability analysis for a broad class of boundary conditions in the case when the relaxation system is 2 x 2.

Shock reflection for the damped P -system

Abstract: The global existence and the asymptotic behavior of the weak entropy solution, the piecewise smooth solution with one shock discontinuity, on a strip domain is investigated in the present paper. We show that, for small smooth initial data and boundary value with only one small jump at (x, t) = (0, 0), the piecewise smooth solution with one shock discontinuity exists globally in time. The shock discontinuity begins from (x, t) = (0, 0), moves forward and reflects in a finite time at the boundary x = 1 to form a 1-shock, which goes backward and reflects at x = 0 also in a finite time to create a new 2-shock. The shock strength decays exponentially and never disappears in finite time. As t → ∞, this solution converges to a constant state determined by the initial and the boundary conditions.

Forbidden planes for Rayleigh waves

Abstract: Existence of "forbidden" planes, on which Rayleigh waves cannot propagate, is discussed. A mathematical model for anisotropic materials possessing "forbidden" planes is constructed. An example of transversely isotropic material having "forbidden" planes is presented.

Stabilization of elastic plates with variable coefficients and dynamical boundary control

Abstract: The aim of this paper is to investigate the stabilization of a hybrid system composed of a plate equation with variable coefficients and two ordinary differential equations under some suitable feedbacks. A rational energy decay rate is established by the multiplier method and the Riemannian geometry method, and the uniform energy decay rate for a simplified system is obtained.

Existence and uniqueness of solutions of Smoluchowski's coagulation equation with source terms

Abstract: We prove the existence and uniqueness of a solution to the Smoluchowski coagulation equation with source terms. The coagulation equation with source terms is potentially useful in applications because one sometimes tries to control coagulation processes by the introduction of particles of various sizes into the system. The existence proof given here differs in style from most other existence proofs in two respects. First, it is not based on a finite-dimensional truncation of the coagulation equation; and secondly, it is achieved with a weaker hypothesis than is usually assumed on the initial data.

A modified Ginzburg-Landau model for Josephson junctions in a ring

Abstract: We consider the Ginzburg-Landau functional to analyze SNS junctions in a one-dimensional ring. We compare several canonical scalings. The linearized problem is solved to obtain the phase transition curves. We compute the Γ-limit of the functional in the different scalings. The interaction of several junctions is analyzed. We study the zero set of the order parameter for distinguished values of the flux. Finally, we compute the currents in the weakly nonlinear regime.

An analytical study of the Kelvin-Helmholtz instabilities of compressible, magnetized tangential velocity discontinuities with generalized polytrope laws

Abstract: The linear Kelvin-Helmholtz instability of tangential velocity discontinuities in high velocity magnetized plasmas with isotropic or anisotropic pressure is investigated. A new analytical technique applied to the magnetohydrodynamic equations with generalized polytrope laws (for the pressure parallel and perpendicular to the magnetic field) yields the complete structure of the unstable, standing waves in the (inverse plasma beta, Mach number) plane for modes at arbitrary angles to the flow and the magnetic field. The stable regions in the (inverse plasma beta, propagation angle) plane are mapped out via a level curve analysis, thus clarifying the stabilizing effects of both the magnetic field and the compressibility. For polytrope indices corresponding to the double adiabatic and magnetohydrodynamic equations, the results reduce to those obtained earlier using these models. Detailed numerical results are presented for other cases not considered earlier, including the cases of isothermal and mixed waves. Also, for modes propagating along or opposite to the magnetic field direction and at general angles to the flow, a criterion is derived for the absence of standing wave instability—in the isotropic MHD case, this condition corresponds to (plasma beta) < 1.

A geometric evolution problem

Abstract: A traditional approach to compression moulding of polymers involves the study of a generalized Hele-Shaw flow of a power-law fluid, and leads to the p-Poisson equation for the instantaneous pressure in the fluid. By studying the convex dual of an equivalent extremal problem, one may let the power-law index of the fluid tend to zero. The solution of the resulting extremal problem, referred to as the asymptotically dual problem, is known to have the property that the flow is always directed towards the closest point on the boundary.In this paper we use this property to introduce the concept of boundary velocity in the case of piecewise C2 domains with only convex corners, and we also give an explicit solution to the asymptotically dual problem in this case. This involves the study of certain topological properties of the ridge of planar domains.With use of the boundary velocity, we define a geometric evolution problem and the concept of classical solutions of it. We prove a uniqueness theorem and use a comparison principle to study the persistence of corners. we actually estimate "waiting times" for corners, in terms of geometric quantities of the initial domain.

Convergence to stationary states in the Maxwell-Bloch system from nonlinear optics

Abstract: are imposed. The electromagnetic field is governed by the classical Maxwell equations, whereas the polarizable medium occupying the set G is modelled as a gas of quantum mechanical systems with two energy levels as described in [5] and [11]. Here f! C I3 is an arbitrary spatial domain, G C fl a certain subset of fl, and Ti C dfl, '=f Also, the whole space case fi = R3 without boundary condition 1.4 is under consideration. The unknown functions are the electric and magnetic fields E, H, which depend on the time t > 0 and the space variable x 6 Q and the dielectic polarization P defined on the set R+ x G. Furthermore, N denotes the difference of the densities of the electrons in the excited and in the ground state. It is also an unknown function defined on R+ x G. In (1.1) the function P is the extension of P on R x 17 defined by zero on the set R+ x (f1\\G). The physical meaning of the boundary condition 1.4 is that Ti is perfectly conducting, such that the tangential component of the electric field must vanish.

Sufficient condition for the existence of solutions of a free boundary problem

Abstract: In this paper we study the existence of critical points of a functional depending on Ω ⊂ R2 through its perimeter and the solution of the Dirichlet problem in Ω, under the constraint that the measure of Ω is given. We give a sufficient condition for the existence of critical points using the implicit function theorem.

A remark on the existence of global BV solutions for a nonlinear hyperbolic wave equation

Abstract: By means of a suitable change of variables we obtain, by application of a general result by Dafermos and Hsiao, cf. [2], an existence theorem in L°° fl BV\\oc of a weak solution of the system corresponding to the quasilinear hyperbolic equation 0tt p'{x) xx + t + F((f>) = 0 in I x [0, +oo[ , for small initial data in BV. This theorem is a partial extension of Dafermos's result for the case with = 0, proved in [1], 1. The auxiliary system. Let us consider the following Cauchy problem: -p'(x)4>xx +4>t + F(j>) = o , (x,t)e Rx[o,+oo[, (l.i) ~4>{x,Q) = 4>q{x), Mx,0) = v0(x) , xeR, (1.2) where p is a given smooth function such that p'(£) > 0, V( e R, and F : R —> K is a smooth function verifying F(0) = 0. We assume, to simplify, the condition p'(0) = 1. Putting u = (f>x, v = t we can write (1.1), (1.2) as a Cauchy problem for a hyperbolic system: ) = 0 {x, 0) = (j>0(x), u(x, 0) = uq(x) = (j>0x(x), v(x, 0) = vo(x) , (1.4) For technical reasons, if we choose k > 1, we can, by putting u(x, t) = u(kx, kt), v(x, t) = v(kx,kt), (x,t) = \\ (f)(kx, kt), replace (1.3), (1.4) by ( t=v ut=vx (i,t)elx[0,+oo[, (1-3') , vt — p'(u)ux + kv + F() = kF(k), 0) = o{x) = ~4>0{kx) , u(x,0) = u0(x) = uo(kx) = 0x(x) , (1.4') v(x, 0) = vo(x) = vo(kx) , x G R . For to € [1, +oo[, let us introduce a — am = (to2 — 2to + 2)-1/2 , b = bm = (to2 + 2m + 2)-1/2 ,

The Riemann problem for an elastic string with a linear Hooke’s law

Abstract: We solve the Riemann problem for vibrations of an infinite, planar, perfectly elastic string with a linear relation between the string tension and the local stretching, i.e., with a linear Hooke's law. The motion is governed by a 4 x 4 system of conservation laws that is linearly degenerate in all wave families. Conservation of energy leads to L2 estimates that are used in the analysis.