Constitutive Relations for Monatomic Gases¶Based on a¶Generalized Rational Approximation¶to the Sum of the Chapman-Enskog Expansion

. We study the asymptotic behavior as time goes to infinity of solutions to the initial‐boundary‐value problem on the half space for a one‐dimensional model system for the isentropic flow of a compressible viscous gas, the so‐called p‐system with viscosity. As boundary conditions, we prescribe the constant state at infinity and require that the velocity be zero at the boundary . When the velocity at infinity is negative and satisfies a condition on the magnitude, we prove that if the initial data are suitably close to those for the corresponding outgoing viscous shock profile, which is suitably far from the boundary, then a unique solution exists globally in time and tends toward the properly shifted viscous shock profile as the time goes to infinity. The proof is given by an elementary energy method. (Accepted March 2, 1998)     

On the Oppenheimer‐Volkoff Equations in General Relativity

We introduce a new formulation of the Oppenheimer‐Volkoff (O‐V) equations, a system of ordinary differential equations that models the interior of a star in general relativity, and we use this to give a completely rigorous mathematical analysis of solutions. In particular, we prove that, under mild assumptions on the equation of state, black holes never form in solutions of the O‐V equations. As a corollary, this implies that the portion of the empty‐space Schwarzschild solution inside the Schwarzschild radius cannot be obtained as a limit of O‐V solutions having non‐zero density. We also prove that if the density ρ at radius r is ever larger than where M(r) is the total mass inside radius r, then M must become negative for some positive radius. We interpret M<0 as a condition for instability because we show that if the pressure is a decreasing function of r, then M(r)<0 at some r>0 implies that the pressure tends to infinity before r=0. (Accepted October 28, 1996)     

Uniqueness in linear elastostatics for problems involving unbounded bodies

The problem of the uniqueness of elastostatic solutions to various boundary value problems involving unbounded two- and three-dimensional bodies is considered. The general boundary value problem is first considered for anisotropic bodies on which the elasticity tensor is uniformly positive definite. The displacement problem is then considered for the cases where the body is homogeneous and isotropic and the elasticity tensor is strongly-elliptic. Finally, the traction problem on homogeneous, isotropic bodies is considered with fairly little restriction placed on the Lamé moduli. In the first two cases no restriction is placed on the geometry of the body other than the normal assumptions allowing for the use of the divergence theorem. For the traction problem, however, it is assumed that one component of the outward normal to the boudary of the body almost never vanishes. In each case it is shown that the desired uniqueness results hold with fairly mild restrictions placed on the displacement or stress fields (depending on the problem) in a neighborhood of infinity. Where possible, the results are shown to hold even though the solutions may possess the sort of discontinuities and singularities commonly found in problems involving cracks, corners, and bonded interfaces.Related results involving the insolvability of certain non-trivial boundary value problems and the behavior of certain elastic states are, where appropriate, developed.     

Well-Posedness, Instabilities, and Bifurcation Results for the Flow in a Rotating Hele�CShaw Cell

We study the radial movement of an incompressible fluid located in a Hele–Shaw cell rotating at a constant angular velocity in the horizontal plane. Within an analytic framework, local existence and uniqueness of solutions is proved, and it is shown that the unique rotationally invariant equilibrium of the flow is unstable. There are, however, other time-independent solutions: using surface tension as a bifurcation parameter we establish the existence of global bifurcation branches consisting of stationary fingering patterns. The same results can be obtained by fixing the surface tension while varying the angular velocity. Finally, it is shown that the equilibria on a global bifurcation branch converge to a circle as the surface tension tends to infinity, provided they stay suitably bounded.     

Orbits Homoclinic to Exponentially Small Periodic Orbits for a Class of Reversible Systems. Application to Water Waves

In this paper a class of reversible analytic vector fields is investigated near an equilibrium. For these vector fields, the part of the spectrum of the differential at the equilibrium which lies near the imaginary axis comes from the perturbation of a double eigenvalue 0 and two simple eigenvalues , . In the first part of this paper, we study the 4-dimensional problem. The existence of a family of solutions homoclinic to periodic orbits of size less than μ^N for any fixed N, where μ is the bifurcation parameter, is known for vector fields. Using the analyticity of the vector field, we prove here the existence of solutions homoclinic to a periodic orbit the size of which is exponentially small ( of order . This result receives its significance from the still unsolved question of whether there exist solutions that are homoclinic to the equilibrium or whether the amplitudes of the oscillations at infinity have a positive infimum. In the second part of this paper we prove that the exponential estimates still hold in infinite dimensions. This result cannot be simply obtained from the study of the 4-dimensional analysis by a center-manifold reduction since this result is based on analyticity of the vector field. One example of such a vector field in infinite dimensions occurs when describing the irrotational flow of an inviscid fluid layer under the influence of gravity and small surface tension (Bond number ) for a Froude number F close to 1. In this context a homoclinic solution to a periodic orbit is called a generalized solitary wave. Our work shows that there exist generalized solitary waves with exponentially small oscillations at infinity. More precisely, we prove that for each F close enough to 1, there exist two reversible solutions homoclinic to a periodic orbit, the size of which is less than , l being any number between 0 and π and satisfying . (Accepted October 2, 1995)     

Application of Uniform Asymptotics to the Second Painlevé Transcendent

In this work we propose a new method for investigating connection problems for the class of nonlinear second‐order differential equations known as the Painlevé equations. Such problems can be characterized by the question as to how the asymptotic behaviours of solutions are related as the independent variable is allowed to pass towards infinity along different directions in the complex plane. Connection problems have been previously tackled by a variety of methods. Frequently these are based on the ideas of isomonodromic deformation and the matching of WKB solutions. However, the implementation of these methods often tends to be heuristic in nature and so the task of rigorising the process is complicated. The method we propose here develops uniform approximations to solutions. This removes the need to match solutions, is rigorous, and can lead to the solution of connection problems with minimal computational effort. Our method relies on finding uniform approximations of differ ential equations of the generic form as the complex‐valued parameter . The details of the treatment rely heavily on the locations of the zeros of the function F in this limit. If they are isolated, then a uniform approximation to solutions can be derived in terms of Airy functions of suitable argument. On the other hand, if two of the zeros of F coalesce as , then an approximation can be derived in terms of parabolic cylinder functions. In this paper we discuss both cases, but illustrate our technique in action by applying the parabolic cylinder case to the “classical” connection problem associated with the second Painlevé transcendent. Future papers will show how the technique can be applied with very little change to the other Painlevé equations, and to the wider problem of the asymptotic behavio ur of the general solution to any of these equations. (Accepted May 15, 1997)     

Numerical Analysis of Axisymmetric Buckling of Plates under Radial Compression

Nonlinear boundaryvalue problems of axisymmetric buckling of simply supported and clamped plates under radial compression are formulated for a system of six firstorder ordinary differential equations with independent fields of finite displacements and rotations. Multivalued solutions are obtained by the shooting method with specified accuracy. Bifurcation of the solutions of the problem is studied, and a parametric bifurcation diagram is constructed for various values of the loading parameter. Curves of buckling modes are given for three branches of the solution. The numerical results agree with available theoretical data.     

Exact Solution of the Wave‐Resistance Problem for a Submerged Cylinder. I. Linearized Theory

. We consider the problem of finding a holomorphic function in a strip with a cut ${\cal A}= \{(x,y) : \, x\in\RE,\,\,0 satisfying some prescribed linear conditions on the boundary. The problem has a one‐parameter family of solutions in the class of sectionally holomorphic functions in ?, vanishing for $|x|\to\infty. We consider the problem of finding a holomorphic function in a strip with a cut satisfying some prescribed linear conditions on the boundary. The problem has a one‐parameter family of solutions in the class of sectionally holomorphic functions in ?, vanishing for . A special solution can be selected by fixing the value of the circulation around the cut. The problem is obtained by linearization of the equations of the wave‐resistance problem for a “slender” cylinder submerged in a heavy fluid and moving at uniform speed in the direction orthogonal to its generators. The results obtained, besides their own interest, are a crucial step for the resolution of the non‐linear problem. (Accepted October 14, 1998)     

Axisymmetric Solutions of the Euler Equations for Polytropic Gases

We construct rigorously a three‐parameter family of self‐similar, globally bounded, and continuous weak solutions in two space dimensions for all positive time to the Euler equations with axisymmetry for polytropic gases with a quadratic pressure‐density law. We use the axisymmetry and self‐similarity assumptions to reduce the equations to a system of three ordinary differential equations, from which we obtain detailed structures of solutions besides their existence. These solutions exhibit familiar structures seen in hurricanes and tornadoes. They all have finite local energy and vorticity with well‐defined initial and boundary values. These solutions include the one‐parameter family of explicit solutions reported in a recent article of ours. (Accepted October 29, 1996)     

Traveling gravity water waves in two and three dimensions

This paper discusses the bifurcation theory for the equations for traveling surface water waves, based on the formulation of Zakharov [58] and of Craig and Sulem [15] in terms of integro-differential equations on the free surface. This theory recovers the well-known picture of bifurcation curves of Stokes progressive wavetrains in two-dimensions, with the bifurcation parameter being the phase velocity of the solution. In three dimensions the phase velocity is a two-dimensional vector, and the resulting bifurcation equations describe two-dimensional bifurcation surfaces, with multiple intersections at simple bifurcation points. The integro-differential formulation on the free surface is posed in terms of the Dirichlet–Neumann operator for the fluid domain. This lends itself naturally to numerical computations through the fast Fourier transform and surface spectral methods, which has been implemented in Nicholls [32]. We present a perturbation analysis of the resulting bifurcation surfaces for the three-dimensional problem, some analytic results for these bifurcation problems, and numerical solutions of the surface water waves problem, based on a numerical continuation method which uses the spectral formulation of the problem in surface variables. Our numerical results address the problem in both two and three dimensions, and for both the shallow and deep water cases. In particular we describe the formation of steep hexagonal traveling wave patterns in the three-dimensional shallow water regime, and their transition to rolling waves, on high aspect ratio rectangular patterns as the depth increases to infinity.     

On the Initial‐Boundary‐Value Problem for the Linearized Equations of Magnetohydrodynamics

We study the initial‐boundary‐value problem for the linearized equations of ideal magnetohydrodynamics defined on a bounded domain whose boundary is a regular magnetic surface. We show that it is well‐posed in and that the loss of regularity of solutions always occurs for some initial data in (Accepted October 23, 1997)     

Hopf Bifurcation on a Two-Neuron System with Distributed Delays: A Frequency Domain Approach

In this paper, a more general two-neuron model with distributed delays and weak kernel is investigated. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of bifurcation parameter point is determined. Furthermore, we found that if the mean delay is used as a bifurcation parameter, Hopf bifurcation occurs for the weak kernel. This means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value. The direction and stability of the bifurcating periodic solutions are determine by the Nyquist criterion and the graphical Hopf bifurcation theorem. Some numerical simulations for justifying the theoretical analysis are also given.     

Quasi‐One‐Dimensional Riemann Problems and Their Role in Self‐Similar Two‐Dimensional Problems

We study two‐dimensional Riemann problems with piecewise constant data. We identify a class of two‐dimensional systems, including many standard equations of compressible flow, which are simplified by a transformation to similarity variables. For equations in this class, a two‐dimensional Riemann problem with sectorially constant data becomes a boundary‐value problem in the finite plane. For data leading to shock interactions, this problem separates into two parts: a quasi‐one‐dimensional problem in supersonic regions, and an equation of mixed type in subsonic regions. We prove a theorem on local existence of solutions of quasi‐one‐dimensional Riemann problems. For 2 × 2 systems, we generalize a theorem of Courant & Friedrichs, that any hyperbolic state adjacent to a constant state must be a simple wave. In the subsonic regions, where the governing equation is of mixed hyperbolic‐elliptic type, we show that the elliptic part is degenerate at the boundary, with a nonlinear variant of a degeneracy first described by Keldysh. (Accepted December 4, 1997)     

Normal Compliance Contact Models with Finite Interpenetration

We study contact problems with contact models of normal compliance type, where the compliance function tends to infinity for a given finite interpenetration. Such models are physically more realistic than standard normal compliance models, where the interpenetration is not restricted, because the interpenetration is typically justified by deformations of microscopic asperities of the surface; therefore it should not exceed a certain value that corresponds to a complete flattening of the asperities. The model can be interpreted as intermediate between the usual normal compliance models and the unilateral contact condition of Signorini type. For the static problem without friction, we prove the existence and uniqueness of solutions and establish the equivalence to an optimization problem. For the static problem with Coulomb friction, we show the existence of a solution. The analysis is based on an approximation of the problems by standard normal compliance models, the derivation of regularity results for these auxiliary problems in Sobolev spaces of fractional order by a special translation technique, and suitable limit procedures.     

Bifurcation analysis of a tri-neuron neural network model in the frequency domain

In this paper, a class of neural network models with three neurons is considered. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of the bifurcation parameter point is determined. If the coefficient μ is chosen as a bifurcation parameter, it is found that Hopf bifurcation occurs when the parameter μ passes through a critical value. The direction and the stability of Hopf bifurcation periodic solutions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem. Some numerical simulations for justifying the theoretical analysis are also provided.     

Computational analysis on Hopf bifurcation and stability for a consumer–resource model with nonlinear functional response

We consider a consumer–resource model with nonlinear functional response and reaction–diffusion terms. By taking the growth rate of the resource as the parameter, we give a computational and theoretical analysis on Hopf bifurcation emitting from the positive equilibrium for the model and discuss the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solutions by space decomposition and vector operation techniques. It is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Some numerical examples are presented to support and illustrate our theoretical analysis.     

Viscosity Solutions of a Degenerate Parabolic-Elliptic System Arising in the Mean-Field Theory of Superconductivity

In a Type‐II superconductor the magnetic field penetrates the superconducting body through the formation of vortices. In an extreme Type‐II superconductor these vortices reduce to line singularities. Because the number of vortices is large it seems feasible to model their evolution by an averaged problem, known as the mean-field model of superconductivity. We assume that the evolution law of an individual vortex, which underlies the averaging process, involves the current of the generated magnetic field as well as the curvature vector. In the present paper we study a two‐dimensional reduction, assuming all vortices to be perpendicular to a given direction. Since both the magnetic field H and the averaged vorticity ω are curl‐free, we may represent them via a scalar magnetic potential q and a scalar stream function ψ, respectively. We study existence, uniqueness and asymptotic behaviour of solutions (ψ, q) of the resulting degenerate elliptic‐parabolic system (with curvature taken into account or not) by means of viscosity and weak solutions. In addition we relate (ψ, q) to solutions (ω, H) of the mean‐field equations without curvature. Finally we construct special solutions of the corresponding stationary equations with two or more superconducting phases. (Accepted August 8, 1997)     

Rigorous Results in Steady Finger Selection in Viscous Fingering

This paper concerns the existence of a steadily translating finger solution in a Hele-Shaw cell for small but non-zero surface tension (ɛ²). Though there are numerous numerical and formal asymptotic results for this problem, we know of no mathematically rigorous results that address the selection problem. We rigorously conclude that for relative finger width λ in the range , with small, analytic symmetric finger solutions exist in the asymptotic limit of surface tension if and only if the Stokes constant for a relatively simple nonlinear differential equation is zero. This Stokes constant S depends on the parameter and earlier calculations by a number of authors have shown it to be zero for a discrete set of values of a. The methodology consists of proving the existence and uniqueness of analytic solutions for a weak half-strip problem for any λ in a compact subset of (0, 1). The weak problem is shown to be equivalent to the original finger problem in the function space considered, provided we invoke a symmetry condition. Next, we consider the behavior of the solution in a neighborhood of an appropriate complex turning point for the restricted case , for some . This turning point accounts for exponentially small terms in ɛ, as ɛ→0⁺ that generally violate the symmetry condition. We prove that the symmetry condition is satisfied for small ɛ when the parameter a is constrained appropriately. (Accepted July 4, 2002 Published online January 15, 2003) Communicated by F. OTTO     

Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system

In this paper, we address the problem of the bifurcation control of a delayed fractional-order dual model of congestion control algorithms. A fractional-order proportional–derivative (PD) feedback controller is designed to control the bifurcation generated by the delayed fractional-order congestion control model. By choosing the communication delay as the bifurcation parameter, the issues of the stability and bifurcations for the controlled fractional-order model are studied. Applying the stability theorem of fractional-order systems, we obtain some conditions for the stability of the equilibrium and the Hopf bifurcation. Additionally, the critical value of time delay is figured out, where a Hopf bifurcation occurs and a family of oscillations bifurcate from the equilibrium. It is also shown that the onset of the bifurcation can be postponed or advanced by selecting proper control parameters in the fractional-order PD controller. Finally, numerical simulations are given to validate the main results and the effectiveness of the control strategy.

     

Stable Nucleation for the Ginzburg-Landau System with an Applied Magnetic Field

We consider the Ginzburg‐Landau system with an applied magnetic field and analyze the behavior of solutions when the domain is a cylinder (of radius ) and the applied field is parallel to the axis. It is shown that there is an upper critical value such that if the modulus of the applied field is greater than , the normal (nonsuperconducting) state (in which the order parameter is identically zero) is stable and if the modulus of the applied field is slightly below , the normal state is unstable. In addition, it is shown that there is a positive lower critical value such that the normal state is unstable if the modulus of the applied field is less than and stable if the modulus is slightly above . In the case of type‐II materials for whic h the Ginzburg‐Landau constant κ is large, it is shown that there is a discrete set of radii ℬ(κ) such that if and is sufficiently large, then for each applied field of modulus slightly less than (or slightly more than ) there is precisely one small superconducting solution (up to a gauge transformation) which is stable. Moreover for this solution, the complex‐valued order parameter ψ is zero only on the axis of the cylinder, and its winding number is proportional to the product of κ² and the cross‐sectional area of the cylinder. In addition, the solution exhibits “surface superconductivity” as predicted by the physicists de Gennes and St. James. (Accepted July 15, 1996)