Showing papers in "Quarterly of Applied Mathematics in 2006"

Viscous flow due to a shrinking sheet

Abstract: The viscous flow induced by a shrinking sheet is studied. Existence and (non)uniqueness are proved. Exact solutions, both numerical and in closed form, are found.

Exponential stability in linear viscoelasticity

Abstract: We address the study of the asymptotic behavior of solutions to an abstract integrodifferential equation modeling linear viscoelasticity. Framing the equation in the past history setting, we analyze the exponential stability of the related semigroup S(t) with dependence on the convolution kernel, providing a more general sufficient condition than the usual one present in the literature.

When is a linear system conservative

Abstract: We consider infinite-dimensional linear systems without a-priori well-posedness assumptions, in a framework based on the works of M. Livsic, M. S. Brodskiǐ, Y. L. Smuljan, and others. We define the energy in the system as the norm of the state squared (other, possibly indefinite quadratic forms will also be considered). We derive a number of equivalent conditions for a linear system to be energy preserving and hence, in particular, well posed. Similarly, we derive equivalent conditions for a system to be conservative, which means that both the system and its dual are energy preserving. For systems whose control operator is one-to-one and whose observation operator has dense range, the equivalent conditions for being conservative become simpler, and reduce to three algebraic equations.

Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters

Abstract: : We consider electromagnetic interrogation problems for complex materials involving distributions of polarization mechanisms and also distributions for the parameters in these mechanisms. a theoretical and computational framework for such problems is given. Computational results for specific problems with multiple Debye mechanisms are given in the case of discrete, uniform, log-normal, and log-Bi-Gaussian distributions.

On stability of linear time-varying second-order differential equations

Abstract: We derive sufficient conditions for stability and asymptotic stability of second order, scalar differential equations with differentiable coefficients.

Peristaltic transport of a Herschel-Bulkley fluid in contact with a Newtonian fluid

Abstract: Peristaltic transport of Herschel-Bulkley fluid in contact with a Newtonian fluid in a channel is investigated for its various applications to flows with physiological fluids (blood, chyme, intrauterine fluid, etc.). The primary application is when blood flows through small vessels; blood has a peripheral layer of plasma and a core region of suspension of all the erythrocytes. That is, in the modeling of blood flow, one needs to consider the core region consisting of a yield stress fluid and the peripheral region consisting of a Newtonian fluid. Peristaltic pumping of a yield stress fluid in contact with a Newtonian fluid has not previously been studied in detail. Our goal is to initiate such a study. The Herschel-Bulkley fluid model considered here reduces to the power law model in the absence of yield stress. The stream function, the velocity field, and the equation of the interface are obtained and discussed. When the yield stress TO → 0 and when the index n = 1, our results agree with those of Brasseur et al. (J. Fluid Mech. 174 (1987), 495) for peristaltic transport of the Newtonian fluid. It is observed that for a given flux Q the pressure rise Ap increases with an increase in the amplitude ratio Φ. Furthermore, the results obtained for the flow characteristics reveal many interesting behaviors that warrant further study of the peristaltic transport models with two immiscible physiological fluids.

SH-waves in laminated plates

Abstract: A mathematical model for analyzing horizontally polarized shear surface (SH) waves propagating in laminated plates is presented. All the layers are assumed to have monoclinic symmetry. Three types of boundary conditions imposed on the outer surfaces of a plate are considered. A variant of the modified transfer matrix (MTM) method is developed. Closed form dispersion relations are obtained for plates consisting of one or two orthotropic layers. Asymptotic solutions for orthotropic two- and three-layered plates are derived.

Convergence to equilibrium for the damped semilinear wave equation with critical exponent and dissipative boundary condition

Abstract: This paper is concerned with the asymptotic behavior of the solution to the following damped semilinear wave equation with critical exponent: u tt + u t - Δu + f(x, u) = 0, (x,t) ∈ Ω × R + (0.1) subject to the dissipative boundary condition ∂ ν u+u+u t = 0, t > 0, x ∈ Γ (0.2) and the initial conditions u| t=0 = u 0 (x), u t | t=0 = u 1 (x), x ∈ Ω, (0.3) where Ω is a bounded domain in R 3 with smooth boundary F, v is the outward normal direction to the boundary, and f is analytic in u. In this paper convergence of the solution to an equilibrium as time goes to infinity is proved. While these types of results are known for the damped semilinear wave equation with interior dissipation and Dirichlet boundary condition, this is, to our knowledge, the first result with dissipative boundary condition and critical growth exponent.

Finite-dimensional attractor for the viscous Cahn-Hilliard equation in an unbounded domain

Abstract: We consider the viscous Cahn-Hilliard equation in an infinite domain. Due to the noncompactness of operators, we use weighted Sobolev spaces to prove that the semigroup generated by this equation has the global attractor which has finite Hausdorff dimension.

decay estimates for the Cauchy problem of linear thermoelastic systems with second sound in one space variable

Abstract: L p - L q decay estimates of solutions to the Cauchy problem of linear thermoelastic systems with second sound in one space variable will be studied in this paper. First, by dividing the frequency of phase space of the Fourier transformation into different regions, the asymptotic behavior of characteristic roots of the coefficient matrix is obtained by carefully analyzing the effect of the different regions. Second, with the help of the information on the characteristic roots and by using the interpolation theorem, the L p - L q decay estimate of solutions to the Cauchy problem of the linear thermoelastic system with second sound in one space variable is obtained.

Asymptotic behaviour of solutions of lubrication problem in a thin domain with a rough boundary and Tresca fluid-solid interface law

Abstract: We study the asymptotic behavior of the solution of a Stokes flow in a thin domain, with a thickness of order e, and a rough surface The roughness is defined by a quasi-periodic function with period e We suppose that the flow is subject to a Tresca fluid-solid interface condition We prove a new result on the lower-semicontinuity for the two-scale convergence, which allows us to obtain rigorously the limit problem and to establish the uniqueness of its solution

Homogenization of stratified thermoviscoplastic materials

Abstract: In the present paper we study the homogenization of the system of partial differential equations posed in a < x < b, 0 < t < T, completed by boundary conditions on v e and by initial conditions on v e and θ e . The unknowns are the velocity v e and the temperature θ e , while the coefficients ρ e , μ e and c e are data which are assumed to satisfy 0 < c 1 ≤ μ e (x,s) ≤ c 2 , 0 < c 3 ≤ c e (x,s) ≤ c 4 , 0 < c 5 ≤ ρ e (x) ≤ c 6 , -c 7 < ∂μ e ∂s(x, S) < 0, |c e (x, s) - c e (x, s')| < ω(|s - s'|). This sequence of one-dimensional systems is a model for the homogenization of nonhomogeneous, stratified, thermoviscoplastic materials exhibiting thermal softening and a temperature-dependent rate of plastic work converted into heat. Under the above hypotheses we prove that this system is stable by homogenization. More precisely one can extract a subsequence e' for which the velocity v e' and the temperature θ e converge to some homogenized velocity v° and some homogenized temperature θ° which solve a system similar to the system solved by u e and θ e , for coefficients p°, μ° and c° which satisfy hypotheses similar to the hypotheses satisfied by ρ e , μ e and c e . These homogenized coefficients ρ°, μ° and c° are given by some explicit (even if sophisticated) formulas. In particular, the homogenized heat coefficient c° in general depends on the temperature even if the heterogeneous heat coefficients c e do not depend on it.

Finite-energy solutions of mixed 3d div-curl systems

Abstract: This paper describes the existence and representation of certain finite energy (L 2 -)solutions of weighted div-curl systems on bounded 3D regions with C 2 -boundaries and mixed boundary data. Necessary compatibility conditions on the data for the existence of solutions are described. Subject to natural integrability assumptions on the data, it is then shown that there exist L 2 -solutions whenever these compatibility conditions hold. The existence results are proved by using a weighted orthogonal decomposition theorem for L 2 -vector fields in terms of scalar and vector potentials. This representation theorem generalizes the classical Hodge-Weyl decomposition. With this special choice of the potentials, the mixed div-curl problem decouples into separate problems for the scalar and vector potentials. Variational principles for the solutions of these problems are described. Existence theorems, and some estimates, for the solutions of these variational principles are obtained. The unique solution of the mixed system that is orthogonal to the null space of the problem is found and the space of all solutions is described. The second part of the paper treats issues concerning the non-uniqueness of solutions of this problem. Under additional assumptions, this space is shown to be finite dimensional and a lower bound on the dimension is described. Criteria that prescribe the harmonic component of the solution are investigated. Extra conditions that determine a well-posed problem for this system on a simply connected region are given. A number of conjectures regarding the results for bounded regions with handles are stated.

Thin elastic films : The impact of higher order perturbations

Abstract: The asymptotic behavior of an elastic thin film penalized by a van der Wals type interfacial energy is investigated when both its thickness and the magnitude of the additional energy vanish in the limit. Keeping track of both mid-plane and out-of-plane deformations (through the introduction of the Cosserat vector), the resulting behavior strongly depends upon the ratio between thickness and interfacial energy.

On a two-point free boundary problem

Abstract: We study a two-point free boundary problem for a quasilinear parabolic equation. This problem arises in the model of flame propagation in combustion theory. It also arises in the study of the motion of interface moving with curvature in which the studied problem is confined in the conical region bounded by two straight lines and the interface has prescribed touching angles with these two straight lines. Depending on these two touching angles, there are three different cases, namely, area-expanding, area-preserving, and area-shrinking cases. We first give a proof of the global existence in the expanding and preserving cases. Then the convergence to a line in the preserving case is derived. Finally, in the shrinking case, we show the finite-time vanishing and the convergence of the solution to a self-similar solution.

Ray methods for Free Boundary Problems

Abstract: We discuss the use of the WKB ansatz in a variety of parabolic problems involving a small parameter. We analyse the Stefan problem for small latent heat, the Black--Scholes problem for an American put option, and some nonlinear diffusion equations, in each case constructing an asymptotic solution by the use of ray methods.

Scattering relations for point-generated dyadic fields in two-dimensional linear elasticity

Abstract: The problem of scattering of elastic waves by a bounded obstacle in two-dimensional linear elasticity is considered. The scattering problems are presented in a dyadic form. An incident dyadic field generated by a point source is disturbed by a rigid body, a cavity, or a penetrable obstacle. General scattering theorems are proved, relating the far-field patterns due to scattering of waves from a point source set up in either of two different locations. The most general reciprocity theorem is established, and mixed scattering relations are also proved. Finally, a relation between the incident and the scattered wave which refers to the mechanism of energy transfer of the scatterer, the so-called optical theorem, is established.

A way to compute the gravitational potential for near-spherical geometries

Abstract: We use a boundary perturbation technique based on the calculus of moving surfaces to compute the gravitational potential for near-spherical geometries with piecewise constant densities. The perturbation analysis is carried out to third order in the small parameter. The presented technique can be adapted to a broad range of potential problems including geometries with variable densities and surface density distributions that arise in electrostatics. The technique is applicable to arbitrary small perturbations of a spherically symmetric configuration and, in principle, to arbitrary initial domains. However, the Laplace equation for an arbitrary domain can usually be solved only numerically. We therefore concentrate on spherical domains which yield a number of geophysical applications. As an illustration, we apply our analysis to the case of a near spherical triaxial ellipsoid and show that third order estimates for ellipticities such as that of the Earth are accurate to ten digits. We include an appendix that contains a concise, but complete, exposition of the tensor calculus of moving interfaces.

Hilbert formulas for $r$-analytic functions and the Stokes flow about a biconvex lens

Abstract: The so-called r-analytic functions are a subclass of p-analytic functions and are defined by the generalized Cauchy-Riemann system with p(r, z) = r. In the system of toroidal coordinates, the real and imaginary parts of an r-analytic function are represented by Mehler-Fock integrals with densities, which are assumed to be meromorphic functions. Hilbert formulas, establishing relationships between those functions, are derived for the domain exterior to the contour of a biconvex lens in the meridional cross-section plane. The derivation extends the framework of the theory of Riemann boundary-value problems, suggested in our previous work, to solving the three-contour problem for the case of meromorphic functions with a finite number of simple poles. For numerical calculations, Mehler-Fock integrals with Hilbert formulas reduce to the form of regular integrals. The 3D problem of the axially symmetric steady motion of a rigid biconvex lens-shaped body in a Stokes fluid is solved, and the Hilbert formula for the real part of an r-analytic function is used to express the pressure in the fluid via the vorticity analytically. As an illustration, streamlines and isobars about the body, the vorticity and pressure at the contour of the body and the drag force exerted on the body by the fluid are calculated.

The exponential decay of global solutions to the generalized Landau equation near Maxwellians

Abstract: Global-in-time classical solutions near Maxwellians are constructed for the generalized Landau equation in a periodic box for γ > -2. The exponential decay of such a solution is also obtained.

Spatial behavior for a fourth-order dispersive equation

Abstract: In this note we investigate the spatial behavior of a linear equation of fourth order which models several mechanical situations when dispersive and dissipative effects are taken into account. In particular, this equation models the extensional vibration of a bar when we assume that external friction, with a rough substrate for example, is present. We show that for such an equation a Phragmen-Lindelof alternative of exponential type can be obtained. A bound for the amplitude term in terms of boundary data is obtained. Moreover, when friction is absent, we obtain exponential decay results in the case of harmonic vibrations and we prove a polynomial decay estimate for general solutions.

Global solutions of the mean–field, very high temperature Caldeira–Leggett master equation

Abstract: In this paper, global well-posedness as well as regularity of very high temperature Caldeira-Leggett models with repulsive Poisson coupling are proved by using Green function techniques and Fokker-Planck smoothing arguments along with kinetic energy and elliptic estimates.

Parabolic-hyperbolic time-dependent ginzburg-landau-maxwell equations

Abstract: This article is devoted to the long-term dynamics of a parabolic-hyperbolic system arising in superconductivity. In the literature, the existence and uniqueness of the solution have been investigated but, to our knowledge, no asymptotic result is available. For the bidimensional model we prove that the system generates a dissipative semigroup in a proper phase-space where it possesses a (regular) global attractor. Then, we show the existence of an exponential attractor whose basin of attraction coincides with the whole phase-space. Thus, in particular, this exponential attractor contains the global attractor which, as a consequence, is of finite fractal dimension.

Asymmetric equilibrium configurations of hyperelastic cylindrical bodies under symmetric dead loads

Abstract: Homogeneous deformations provided by the nonlinear equilibrium problem of symmetrically loaded isotropic hyperelastic cylindrical bodies are investigated. Depending on the form of the stored energy function, the problem considered may admit asymmetric solutions, besides the expected symmetric solutions. For general compressible materials, the mathematical condition allowing the assessment of these asymmetric solutions, which describe the global path of equilibrium branches, is given. Explicit expressions for evaluating critical loads and bifurcation points are derived. Results and basic relations obtained for general isotropic materials are then specialized for a Mooney-Rivlin and a neo-Hookean material. A broad numerical analysis is performed and the qualitatively more interesting asymmetric equilibrium branches are shown. The influence of the constitutive parameters is discussed, and, using the energy criterion, a number of considerations are carried out concerning the stability of the equilibrium solutions.

Competitive systems with migration and the Poincaré-Bendixson theorem for a 4-dimensional case

Abstract: In this paper, dynamics of the n-species competitive system with migration is studied. It is proved that if the Jacobian matrix of the system is irreducible at every point in Int R 2n + , then there is a defined countable family of invariant (2n - 1)-cells which attract all nonconvergent persistent trajectories. Moreover, it is proved that the Poincare-Bendixson theorem holds for 2-species competitive systems with migration.

Multiplicity of asymmetric solutions for nonlinear elliptic problems

Abstract: In this paper we study the existence of multiple asymmetric positive solutions for the following symmetric problem: {-Δu + (λ - h(x))u = (1 - f(x))u p , x ∈ R N , u(x) > 0, x ∈ R N , u ∈ H 1 (R N ), where λ > 0 is a parameter, h(x) and f(x) are nonnegative radially symmetric functions in L ∞ (R N ), h(x) and f(x) have compact support in R N , f(x) ≤ 1 for all x ∈ R N , 1 < p < +∞ for N = 1,2, 1 < p < N+2/N-2 for N ≥ 3. We prove that for any k = 1,2,..., if λ is large enough the above problem has positive solutions u λ concentrating at k distinct points away from the origin as λ goes to ∞.

Some remarks on homogenization and exact boundary controllability for the one-dimensional wave equation

Abstract: This paper contains three results concerning the homogenization and exact controllability for the one-dimensional wave equation. First, we give sufficient conditions on the initial data to ensure the convergence of the conormal derivatives associated with the wave equation with a rapidly oscillating coefficient and zero Dirichlet boundary conditions. Secondly, we apply this result to prove the existence of a class of initial data whose associated boundary controls are uniformly bounded and obtain some information (in particular, its limit behavior) on this class of data. Finally, we prove that all initial data in L 2 x H -1 may be uniformly controlled but at the price of adding an internal feedback control in our system. The main advantage of this last procedure is that we have explicit formulae for both states and controls.

Homoclinic orbits in predator-prey systems with a nonsmooth prey growth rate

Abstract: This paper deals with Gause-type predator-prey models with a nonsmooth prey growth rate. Our models have a unique positive equilibrium and are under the influence of an Allee effect. A necessary and sufficient condition is given for the existence of homoclinic orbits whose α- and ω-limit sets are the positive equilibrium. The argument used here is based on some results of a system of Lienard type. The relation between homoclinic orbits and the Allee effect is clarified. A simple example is included to illustrate the main result. Some global phase portraits are also attached.

Existence of optimal shape for a system of conservation laws in a free air-porous domain

Abstract: We consider a shape optimization problem related to a nonlinear system of PDE describing the gas dynamics in a free air-porous domain, including gas concentrations, temperature, velocity and pressure. The velocity and pressure are described by the Stokes and Darcy laws, while concentrations and temperature are given by mass and heat conservation laws. The system represents a simplified dry model of gas dynamics in the channel and graphite diffusive layers of hydrogen fuel cells. The model is coupled with the other part of the domain through some mixed boundary conditions, involving nonlinearities, and pressure boundary conditions. Under some assumptions we prove that the system has a solution and that there exists a channel domain in the class of Lipschitz domains minimizing a certain functional measuring the membrane temperature distribution, total current, water vapor transport and channel inlet/outlet pressure drop.

Elliptic equations with diffusion coefficient vanishing at the boundary: Theoretical and computational aspects

Abstract: A class of degenerate elliptic PDEs is considered. Specifically, it is assumed that the diffusion coefficient vanishes on the boundary of the domain. It is shown that if the diffusion coefficient vanishes fast enough, then the problem has a unique solution in the class of smooth functions even if no boundary conditions are supplied. A numerical method is derived to compute solutions for such degenerate equations. The problem is motivated by a certain approach to the recovery of the phase of a wave from intensity measurements.