Showing papers in "Quarterly of Applied Mathematics in 2003"
TL;DR: In this paper, the existence of weak solutions in Sobolev framework is proved by using some compactification properties deduced from the Poisson equation, and the Cauchy Problem for coupled NavierStokes-Poisson equation is investigated.
Abstract: In this paper we investigate the Cauchy Problem for coupled NavierStokes-Poisson equation. The global existence of weak solutions in Sobolev framework is proved by using some compactification properties deduced from the Poisson equation.
84 citations
TL;DR: In this article, the authors used Cattaneo's law for heat conduction instead of Fourier's law and proved the exponential stability of a purely, but slightly damped, hyperbolic system in the radially symmetric case.
Abstract: We consider thermoelastic systems in two or three space dimensions where thermal disturbances are modeled propagating as wavelike pulses traveling at finite speed. This is done using Cattaneo's law for heat conduction instead of Fourier's law. For Dirichlet type boundary conditions, the exponential stability of the now purely, but slightly damped, hyperbolic system is proved in the radially symmetric case.
72 citations
TL;DR: In this article, a one dimensional model simulating the shear in a two-dimensional body was studied and the authors deduced the continuum limit of the model as the lattice parameter goes to zero.
Abstract: In this paper we study a one dimensional model simulating the shear in a two dimensional body. We analyse the discrete system and we deduce the continuum limit of the lattice model as the lattice parameter goes to zero. Different energies are introduced and linked together.
34 citations
TL;DR: In this article, the authors derived an asymptotic model for weakly nonlinear Rayleigh wave propagation on a tangential discontinuity in incompressible magnetohydrodynamics.
Abstract: We derive an asymptotic equation that describes the propagation of weakly nonlinear surface waves on a tangential discontinuity in incompressible magnetohydrodynamics. The equation is similar to, but simpler than, previously derived asymptotic equations for weakly nonlinear Rayleigh waves in elasticity, and is identical to a model equation for nonlinear Rayleigh waves proposed by Hamilton et al. The most interesting feature of the surface waves is that their nonlinear self-interaction is nonlocal. As a result of this nonlocal nonlinearity, smooth solutions break down in finite time, and appear to form cusps.
33 citations
TL;DR: In this article, the existence of global weak solutions to the Navier-Stokes equations for a one-dimensional viscous polytropic ideal gas was proved, and it was shown that neither vacuum states nor concentration states can form and the temperature remains positive in finite time.
Abstract: We prove the existence of global weak solutions to the Navier-Stokes equations for a one-dimensional viscous polytropic ideal gas. We require only that the initial density is in L°°r\\Lfoc with positive infimum, the initial velocity is in Lfoc, and the initial temperature is in Lloc with positive infimum. The initial density and the initial velocity may have differing constant states at x = ±00. In particular, piecewise constant data with arbitrary large jump discontinuities are included. Our results show that neither vacuum states nor concentration states can form and the temperature remains positive in finite time.
31 citations
TL;DR: In this paper, a generalized Moreau-Yosida based primal-dual active set algorithm for the solution of a representative class of bilaterally control constrained optimal control problems with boundary control is developed.
Abstract: A generalized Moreau-Yosida based primal-dual active set algorithm for the solution of a representative class of bilaterally control constrained optimal control problems with boundary control is developed. The use of the generalized Moreau-Yosida approximation allows an efficient identification of the active and inactive sets at each iteration level. The method requires no step-size strategy and exhibits a finite termination property for the discretized problem class. In infinite as well as in finite dimensions a convergence analysis based on an augmented Lagrangian merit function is given. In a series of numerical tests the efficiency of the new algorithm is emphasized.
27 citations
25 citations
TL;DR: In this article, the mathematical formulation of a dissipative Fremond model for shape memory alloys is given in terms of an initial and boundary values problem, and the uniqueness of sufficiently regular solutions is proved by use of a contracting estimates procedure in the case when quadratic dissipative contributions are neglected.
Abstract: The mathematical formulation of a dissipative Fremond model for shape memory alloys is given in terms of an initial and boundary values problem. Uniqueness of sufficiently regular solutions is proved by use of a contracting estimates procedure in the case when quadratic dissipative contributions are neglected in the energy balance. The related existence result is only established while its proof will be detailed by the author in a subsequent paper.
23 citations
23 citations
TL;DR: In this article, countable spectra of different asymptotic patterns of selfsimilar and approximate self-similar types for global and blow-up solutions for the semilinear wave equation were constructed in different ranges of exponent p and dimension N.
Abstract: We construct countable spectra of different asymptotic patterns of selfsimilar and approximate self-similar types for global and blow-up solutions for the semilinear wave equation utt = Au + \\u\\p~1u, x € Rw, t > 0, in different ranges of exponent p and dimension N.
23 citations
TL;DR: In this article, the authors extend the linear stick-slip models of Doi-Edwards and Johnson-Stacer to nonlinear tube reptation models, and show that such models, when combined with probabilistic formulations allowing distributions of relaxation times, provide a good description of dynamic experiments with highly filled rubber in tensile deformations.
Abstract: : We extend the linear stick-slip models of Doi-Edwards and Johnson-Stacer to nonlinear tube reptation models. We then show that such models, when combined with probabilistic formulations allowing distributions of relaxation times, provide a good description of dynamic experiments with highly filled rubber in tensile deformations. A connection to other applications including dielectric polarization and reptation in other viscoelastic materials (e.g., living tissue) is noted.
TL;DR: In this paper, a collisionless Boltzmann equation subject to a periodic array of localized scatters modeling the periodic heterogeneities of the material is derived for semiconductor superlattices.
Abstract: In this paper, we rigorously derive a diffusion model for semiconductor superlattices, starting from a kinetic description of electron transport at the microscopic scale. Electron transport in the superlattice is modelled by a collisionless Boltzmann equation subject to a periodic array of localized scatters modeling the periodic heterogeneities of the material. The limit of a large number of periodicity cells combined with a large-time asymptotics leads to a homogenized diffusion equation which belongs to the class of so-called \"SHE\" models (for Spherical Harmonics Expansion). The rigorous convergence proof relies on fine estimates on the operator modeling the localized scatters.
TL;DR: In this article, it was proved that for some cubic crystals and the directions of elastic symmetry there arise exponentially attenuating with depth surface waves of the non-Rayleigh type, and the existence of forbidden planes for some transversely isotropic half spaces upon which the genuine Rayleigh waves cannot propagate was established.
Abstract: In a previous publication [1] existence of “forbidden” planes for some transversely isotropic half spaces upon which the genuine Rayleigh waves cannot propagate was established. Now, it is proved that for some cubic crystals and the directions of elastic symmetry there arise exponentially attenuating with depth surface waves of the non-Rayleigh type.
TL;DR: In this paper, the authors examined the spectrum of shock profiles for the Jin-Xin relaxation scheme for systems of hyperbolic conservation laws in one spatial dimension and proved that these shock profiles exhibit strong spectral stability in the weak shock limit.
Abstract: We examine the spectrum of shock profiles for the Jin-Xin relaxation scheme for systems of hyperbolic conservation laws in one spatial dimension. By using a weighted norm estimate, we prove that these shock profiles exhibit strong spectral stability in the weak shock limit.
TL;DR: A detailed analysis of the Geselowitz formula for the magnetic induction and for the electric potential fields, due to a localized dipole current density, is provided in this article, where it is shown that the volume integral, which describes the contribution of the conductive tissue to the magnetic field, exhibits a hyper-singular behaviour at the point where the dipole source is located.
Abstract: A detailed analysis of the Geselowitz formula for the magnetic induction and for the electric potential fields, due to a localized dipole current density, is provided. It is shown that the volume integral, which describes the contribution of the conductive tissue to the magnetic field, exhibits a hyper-singular behaviour at the point where the dipole source is located. This singularity is handled both via local regularization of the volume integral as well as through calculation of the total flux it generates. The analysis reveals that the contribution of the primary dipole to the volume integral is equal to the one third of the magnetic field generated by the primary dipole while the rest is due to the distributed conductive tissue surrounding the singularity. Furthermore, multipole expansion is introduced, which expresses the magnetic field in terms of polyadic moments of the electric potential over the surface of the conductor.
TL;DR: In this article, the degenerate semilinear parabolic first initial-boundary value problem is studied, where the problem has a unique solution before a blowup occurs, and the blowup set consists of the single point b. A lower bound and an upper bound of the blow-up time are also given.
Abstract: Let q be a nonnegative real number, and T be a positive real number. This article studies the following degenerate semilinear parabolic first initial-boundary value problem: xqut(x,t) — uxx(x,t) = a25(x — b)f(u{x,t)) for 0 < x < 1,0 < t < T, u(x, 0) = i/j(x) for 0 < x < 1, w(0, t) = u(l, t) = 0 for 0 < t < T, where S(x) is the Dirac delta function, and / and ip are given functions. It is shown that the problem has a unique solution before a blow-up occurs, u blows up in a finite time, and the blow-up set consists of the single point b. A lower bound and an upper bound of the blow-up time are also given. To illustrate our main results, an example is given. A computational method is also given to determine the finite blow-up time.
TL;DR: In this paper, an elastic dissipation model for a cantilevered beam has been studied for determining the relationship between the damping rates and the frequencies using a recently developed, adapted form of the method of separation of variables.
Abstract: In this paper we will study an elastic dissipation model for a cantilevered beam. This problem for a cantilevered beam has been formulated by D.L. Russell as an open problem in [1, 2]. To determine the relationship between the damping rates and the frequencies we will use a recently developed, adapted form of the method of separation of variables. It will be shown that the dissipation model as proposed by D.L. Russell for the cantilevered beam will not always generate damping. Moreover, it will be shown that some solutions can become unbounded.
TL;DR: In this article, the eigenvalue problem for the linear stability of Couette flow between two rotating concentric cylinders to axisymmetric disturbances is considered and it is proved that the principle of exchange of stabilities holds when the cylinders rotate in the same direction and the circulation decreases outwards.
Abstract: The eigenvalue problem for the linear stability of Couette flow between two rotating concentric cylinders to axisymmetric disturbances is considered. It is proved that the principle of exchange of stabilities holds when the cylinders rotate in the same direction and the circulation decreases outwards. The proof is based on the notion of a positive operator which is analogous to a positive matrix. Such operators have a spectral property which implies the principle of exchange of stabilities.
TL;DR: In this article, the existence, uniqueness, and regularity of the solutions of a boundary value problem in a strip, which is obtained by linearization of the equations of the wave-resistance problem for a cylinder semisubmerged in a heavy fluid of constant depth H and moving at uniform velocity c in the direction orthogonal to its generators, were discussed.
Abstract: We discuss existence, uniqueness, and regularity of the solutions of a boundary value problem in a strip, which is obtained by linearization of the equations of the wave-resistance problem for a cylinder semisubmerged in a heavy fluid of constant depth H and moving at uniform velocity c in the direction orthogonal to its generators. We show that the problem has a unique solution, rapidly decreasing at infinity, for every c > \\fgHi where g is the acceleration of gravity. For c < \\/gH, we prove unique solvability provided c ^ c^, where Ck is a known sequence monotonically decreasing to zero. In this case, the related flow has in general nontrivial oscillations at infinity downstream. The appearance of the singular values Cfc can be interpreted in terms of a \"nonresonance condition\" between the length of the cylinder's section and the gravitational wave bifurcating from the free parallel flow at the same velocity c.
TL;DR: In this paper, a priori estimates for the deformation of a beam and a plate are provided. But the authors focus on the worst case for the maximum deformation depending on where a load is placed on a beam or a plate.
Abstract: The paper contains a priori estimates for the deformation of plates and beams. In particular we investigate the \"worst cases\" for the maximum deformation depending on where a load is placed on a beam or plate. The methods of proof use rearrangement argument.
TL;DR: In this article, the existence of kernel sections for the process generated by a non-autonomous wave equation with linear memory with nonlinear damping and the nonlinearity has a growing exponent was proved.
Abstract: We prove the existence of kernel sections for the process generated by a non-autonomous wave equation with linear memory when there is nonlinear damping and the nonlinearity has a critically growing exponent; we also obtain a more precise estimate of upper bound of the Hausdorff dimension of the kernel sections. And we point out that in the case of autonomous systems with linear damping, the obtained upper bound of the Hausdorff dimension decreases as the damping grows for suitable large damping.
TL;DR: In this article, the authors studied the spatial behavior of a Mindlin-type thermoelastic plate with a semi-infinite strip and showed that at infinity a sharper spatial decay holds and it is dominated by the thermal characteristics only.
Abstract: Spatial behaviour is studied for the transient solutions in the bending of a Mindlin-type thermoelastic plate. Some appropriate time-weighted line-integral measures are associated with the transient solutions and the spatial estimates are established for these measures describing spatial behaviour results of the Saint-Venant and PhragmenLindelof type. A complete description of the spatial behaviour is obtained by combining the spatial estimates with time-independent and time-dependent decay and growth rates. For a thermoelastic plate whose middle surface is like a semi-infinite strip, it is shown, by means of the maximum principle, that at infinity a sharper spatial decay holds and it is dominated by the thermal characteristics only. Uniqueness results are also established.
TL;DR: In this article, the authors consider the nonlinear degenerate diffusion equation and determine the value of t*, which is called the waiting time, under some condition imposed on the initial function, the interfaces do not move on some time interval [0, t*].
Abstract: We consider the nonlinear degenerate diffusion equation. The most striking manifestation of the nonlinearity and degeneracy is an appearance of interfaces. Under some condition imposed on the initial function, the interfaces do not move on some time interval [0, t*]. In this paper, from numerical points of view, we try to determine the value of t*, which is called the waiting time.
TL;DR: In this article, the authors established some criteria for the positive equilibrium of Eq. (0.1) to be globally asymptotically stable, and then some special cases of Eqs. (1) are investigated further and more global stability results are obtained.
Abstract: Consider the nonautonomous difference equation yn+1= 1 „ = 01 (01) 1 Vn ^ Vn GXp(/3n(l ^i-Vn—i)) where k is a nonnegative integer, ao, cti,..., ak-i are nonnegative constants, a/,is a positive constant and {/3n} is a nonnegative sequence, which is used as a genotype selection model. In this paper, we first establish some criteria for the positive equilibrium of Eq. (0.1) to be globally asymptotically stable. Then some special cases of Eq. (0.1) are investigated further and more global stability results are obtained. Our results also extend and improve some known results in the literature.
TL;DR: In this paper, the authors considered the bistable reaction-diffusion-convection equation dtu + V ■ f(u): = −g(u) + eAu, x G Rn, u £ R (1) is considered.
Abstract: The bistable reaction-diffusion-convection equation dtu + V ■ f(u): = —g(u) + eAu, x G Rn, u £ R (1) is considered. Stationary traveling waves of the above equation are proved to exist when f(u) is symmetric and g(u) is antisymmetric about u = 0. Solutions of initial value problems tend to almost piecewise constant functions within 0{l)e time. The almost constant pieces are separated by sharp interior layers, called fronts. The motion of these fronts is studied by asymptotic expansion. The equation for the motion of the front is obtained. In the case of f = bu2 and g(u) = au(l — u2), where b G RTM and 0 < a G R are constants, the front motion equation takes a more explicit form, showing that the front's speed is V/U _ K+ — ■ T where k is the mean curvature of the front, /i is the width of the planar traveling of (1) in the normal direction n of the front, and T is a vector tangential to the front. Both k and V/x/// • T are elliptic operators, contributing to the shrinkage of closed curves. An ellipse in R2 is found to preserve its shape while shrinking.
TL;DR: In this paper, the authors explored the application of the energy relaxation method to the earlier models to produce solutions free of compressive states and compared with results computed using the standard strain energy for a membrane.
Abstract: During ascent and at other times during flight, the lifting gas of a high altitude balloon is compressed and only able to partially inflate the balloon. In this condition the surface of the balloon will sag to form folds and wrinkles which are difficult to analyze. Previous numerical work to analyze these types of balloons was based on minimizing extrema of potential energy of balloon shapes that included an explicit representation of excess material as folds. These models used the conventional strain energy for linear isotropic membranes and permitted compressive states to enter the solutions. This paper explores the application of the energy relaxation method to the earlier models to produce solutions free of compressive states. Numerical results computed using the relaxed energy are presented and compared with results computed using the standard strain energy for a membrane.
TL;DR: In this article, the vorticity formulation for the lake equations in R2 is studied and the existence and uniqueness of the Cl+a solution to the integral system are established.
Abstract: The vorticity formulation for the lake equations in R2 is studied. We assume that the initial vorticity has the form ui (x, 0) = wo(a:)xn0i where the initial vortex region Q0 is a Cl+a domain and loq 6 Ca (flo) ■ It is shown that the Cauchy problem can be formulated as an integral system. Global existence and uniqueness of the Cl+a solution to the integral system are established. Consequently, the lake equation admits a unique weak solution, global in time, in the form of uj (x, t) = u>t (x) \
t, where Ut (x) e C\" (fit) and d£lt G Ca.
TL;DR: In this article, a new Weber-type transform pair for the representation of a function defined over the domain a < r < oo and which satisfies the Robin mixed boundary condition f{a) + Xf'(a) = 0.
Abstract: In this paper we introduce a new Weber-type transform pair for the representation of a function f(r) defined over the domain a < r < oo and which satisfies the Robin mixed boundary condition f{a) + Xf'(a) = 0. The orthogonality relationships of the transform kernels are derived in both the spatial and the spectral domains as well as Parseval's theorem. We apply this new Weber-type transform pair to solve a mixed boundary value problem in a system of planar layers.
TL;DR: In this paper, the authors apply the Kravtsov-ludwig technique for computing high-frequency fields near cusp caustics and compare these fields, with those predicted by the geometrical optics, for a couple of model problems.
Abstract: It is well known that the usual harmonic ansatz of geometrical optics fails near caustics. However, uniform expansions exist which are valid near and on the caustics, and reduce asymptotically to the usual geometric field far enough from them. In this paper, we apply the Kravtsov-Ludwig technique for computing high-frequency fields near cusp caustics. We compare these fields, with those predicted by the geometrical optics, for a couple of model problems: first, the cusp generated by the evolution of a parabolic initial front in a homogeneous medium, a problem which arises in the highfrequency treatment of cylindrical aberrations, and second, the cusp formed by refraction of the rays emitted from a point source in a stratified medium with a weak interface. It turns out that inside and near the cusp, the geometrical optics solution is significantly different than the Kravtsov-Ludwig solution, but far enough from the caustic that the two solutions are, in fact, in very good agreement.