Static Solutions of the Vlasov-Einstein System

Static solutions of the SO(3)-symmetric Vlasov-Einstein system are studied via a variational approach. For the constitutive relation of the Emden-Fowler type φ(E,F)≡E ^{σ+ 1} F ^k we prove the existence of such solutions of sufficiently small mass-energy, provided 0<σ < k+3/2. These solutions are local minimizers of the energy-Casimir functional, subjected to a variational barrier. Accepted July 16, 2000?Published online January 22, 2001     

Long-time Solvability of the Navier–Stokes Equations in a Rotating Frame with Spatially Almost Periodic Large Data

We consider existence of solutions, for large times, to the Navier–Stokes equations in a rotating frame with spatially almost periodic large data provided by a sufficiently large Coriolis force. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with large-scale phenomena. To show existence of solutions for large times, we use the ℓ ¹-norm of amplitudes. Existence for large times is proven by means of techniques of fast singular oscillating limits and bootstrapping from a global-in-time unique solution to the limit equation.     

Sobolev Periodic Solutions of Nonlinear Wave Equations in Higher Spatial Dimensions

We prove the existence of Cantor families of periodic solutions for nonlinear wave equations in higher spatial dimensions with periodic boundary conditions. We study both forced and autonomous PDEs. In the latter case our theorems generalize previous results of Bourgain to more general nonlinearities of class C ^k and assuming weaker non-resonance conditions. Our solutions have Sobolev regularity both in time and space. The proofs are based on a differentiable Nash–Moser iteration scheme, where it is sufficient to get estimates of interpolation-type for the inverse linearized operators. Our approach works also in presence of very large “clusters of small divisors”.     

Asymptotic Behavior of Global Classical Solutions to the Mixed Initial-Boundary Value Problem for Quasilinear Hyperbolic Systems with Small BV Data

In this paper, we investigate the asymptotic behavior of global classical solutions to the mixed initial-boundary value problem with small BV data for linearly degenerate quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t,x)|t≥0,x≥0}. Based on the existence result on the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of C ¹ traveling wave solutions, provided that the C ¹ norm of the initial and boundary data is bounded and the BV norm of the initial and boundary data is sufficiently small. Applications to quasilinear hyperbolic systems arising in physics and mechanics, particularly to the system describing the motion of the relativistic string in the Minkowski space-time R ¹⁺ⁿ , are also given.     

Asymptotic Analysis for a Mixed Boundary-Value Contact Problem

The variational solution of the nonlinear Signorini contact problem determines also the active contact zone Γ ^c . If the latter is known, then the elastic field is a solution of a linear mixed boundary value problem in which on Γ ^c the normal displacement and tangential traction are given, while on the non-contact part the total traction is zero. Such mixed boundary conditions in general generate singularities of the solution's stress field at the points P ^{( k )} where the boundary conditions change. For smooth data, however, the variational solution of the Signorini contact problem actually belongs to H ²(Ω)², which implies the disappearance of these singularities, i.e., that the corresponding stress intensity factors vanish. This paper is devoted to the characterization of the active contact zone Γ ^c by the vanishing stress intensity factors including their sensitivity with respect to varying Γ ^c for two-dimensional problems provided that Γ ^c consists of a finite number of intervals. We use the method of asymptotic expansions and derive an explicit formula for the sensitivity, which is rigorously justified by employing weighted Sobolev spaces with detached asymptotics. These results can be used to determine the points P ^{( k )} with a corresponding Newton iteration. Accepted July 6, 2000?Published online January 22, 2001     

Experimental Study of Seepage Properties of Broken Sandstone Under Different Porosities

In coal mining the water flow in broken rock is a very common phenomenon. Study of seepage properties of broken rock is one of the basic subjects required in order to understand the stability of rock surrounding roadways, preventing disasters such as water inrush and gas outbursts and developing underground resources. So far, quantitative studies on the nonlinear seepage properties of broken sandstone under different porosities are not extensive in the research literature. In this article, by means of an electro-hydraulic servo-controlled test system (MTS815.02) and a patent seepage device, the seepage properties under different conditions of porosity were tested on broken sandstone of five different grain sizes. Based on the loading method of controlling the axial compression displacement and steady permeating method, we obtained curves of the relation of pore pressure with time, as well as the relation curves between the pore pressure gradient for steady seepage and velocity. Furthermore, we calculated the permeability k and non-Darcy coefficient β corresponding to different porosities by fitting these curves with the binomial expression. This study indicates that: (1) the seepage properties of broken sandstone are closely related to grain size, load levels, and porosity structure; (2) the permeability k decreases, while the coefficient β increases with a decrease in porosity φ, but both the k − φ and the β − φ curves show some local fluctuations; (3) the permeability k of the broken sandstone has a magnitude of 10⁻¹⁴–10⁻¹² m², while the coefficient β ranges from 10¹⁰ to 10¹² m⁻¹. The results obtained provide some information for further study of the nonlinear seepage behavior of broken rock theoretically.     

On the behavior of solutions of systems of linear functional differential equations with constant coefficients and linearly transformed argument in the neighborhood of a singular point

We establish new properties of C ¹(0, +∞)-solutions of systems of linear functional differential equations x′(t) = Ax(t) + Bx(qt) + Cx′(qt) in the neighborhood of the singular point t = 0.     

Unsteady free convection on a vertical cylinder with variable heat and mass flux

The unsteady natural convection boundary layer flow over a semi-infinite vertical cylinder is considered with combined buoyancy force effects, for the situation in which the surface temperature T ^′ _w(x) and C ^′ _w(x) are subjected to the power-law surface heat and mass flux as K(T ^′/r) = −ax ⁿ and D(C ^′/r) = −bx ^m . The governing equations are solved by an implicit finite difference scheme of Crank-Nicolson method. Numerical results are obtained for different values of Prandtl number, Schmidt number ‘n’ and ‘m’. The velocity, temperature and concentration profiles, local and average skin-friction, Nusselt and Sherwood numbers are shown graphically. The local Nusselt and Sherwood number of the present study are compared with the available result and a good agreement is found to exist. Received on 7 July 1998     

On asymptotic properties of solutions of linear differential functional equations with linearly transformed argument

We establish new properties of C ¹[−1, +∞)-solutions of the linear functional differential equation ẋ(t) = ax(t) + bx(qt) + hx(t−1) + cẋ(qt) + rẋ(t−1) in the neighborhood of the singular point t = +∞. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 170–177, April–June, 2006.     

On the Influence of the Kinetic Energy¶on the Stability of Equilibria¶of Natural Lagrangian Systems

Natural Lagrangian systems (T,Π) on R ² described by the equation are considered, where is a positive definite quadratic form in and Π(q) has a critical point at 0. It is constructively proved that there exist a C ^∞ potential energy Π and two C ^∞ kinetic energies T and such that the equilibrium q(t)≡ 0 is stable for the system (T,Π) and unstable for the system . Equivalently, it is established that for C ^∞ natural systems the kinetic energy can influence the stability. In the analytic category this is not true. Accepted: October 20, 1999     

Stabilization using fractional-order PI and PID controllers

This paper presents a solution to the problem of stabilizing a given fractional dynamic system using fractional-order PIλ and PI^λD^μ controllers. It is based on plotting the global stability region in the (k _p, k _i)-plane for the PI^λ controller and in the (k _p , k _i , k _d)-space for the PIλDμ controller. Analytical expressions are derived for the purpose of describing the stability domain boundaries which are described by real root boundary, infinite root boundary and complex root boundary. Thus, the complete set of stabilizing parameters of the fractional-order controller is obtained. The algorithm has a simple and reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of fractional-order control systems.     

Scherk-Type Capillary Graphs

This paper concerns the regularity of a capillary graph (the meniscus profile of liquid in a cylindrical tube) over a corner domain of angle α. By giving an explicit construction of minimal surface solutions previously shown to exist (Indiana Univ. Math. J. 50 (2001), no. 1, 411–441) we clarify two outstanding questions. Solutions are constructed in the case α = π/2 for contact angle data (γ₁, γ₂) = (γ, π − γ) with 0 < γ < π. The solutions given with |γ − π/2| < π/4 are the first known solutions that are not C² up to the corner. This shows that the best known regularity (C^{1, ∈}) is the best possible in some cases. Specific dependence of the H?lder exponent on the contact angle for our examples is given. Solutions with γ = π/4 have continuous, but horizontal, normal vector at the corners in accordance with results of Tam (Pacific J. Math. 124 (1986), 469–482). It is shown that our examples are C^{0, β} up to and including the corner for any β < 1. Solutions with |γ − π/2| > π/4 have a jump discontinuity at the corner. This kind of behavior was suggested by numerical work of Concus and Finn (Microgravity sci. technol. VII/2 (1994), 152–155) and Mittelmann and Zhu (Microgravity sci. technol. IX/1 (1996), 22–27). Our explicit construction, however, allows us to investigate the solutions quantitatively. For example, the trace of these solutions, excluding the jump discontinuity, is C^2/3.     

On {SL(2, \mathbb R)} Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows

Consider the class of C ^r -smooth SL(2, \mathbb R){SL(2, \mathbb R)} valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: |a-\fracp_nq_n| ￡ C exp (-q^r+1+k_n)\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n}), where C >  0 and 0 < k < 1{0 < \kappa <1 } are some constants and \fracP_nq_n{\frac{P_n}{q_n}} is some sequence of irreducible fractions.     

On the <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript> Theory of Hydrostatic Euler Equations

In this paper we study the two-dimensional hydrostatic Euler equations in a periodic channel. We prove the local existence and uniqueness of H ^s solutions under the local Rayleigh condition. This extends Brenier’s (Nonlinearity 12(3):495–512, 1999) existence result by removing an artificial condition and proving uniqueness. In addition, we prove weak–strong uniqueness, mathematical justification of the formal derivation and stability of the hydrostatic Euler equations. These results are based on weighted H ^s a priori estimates, which come from a new type of nonlinear cancellation between velocity and vorticity.     

Characterization of the flow properties of sodium carboxymethylcellulose via mechanical and optical techniques

Sodium carboxymethylcellulose (NaCMC) in solution represents a complex rheological system, since it forms aggregates and associations and hence higher-level structures and, depending on the synthesis, is only found in a molecularly dispersed form in exceptional cases. Rheo-mechanical investigations of the viscoelasticity showed that the Cox-Merz rule is not fulfilled. The aim was therefore to examine whether rheo-optics could be employed to provide more detailed conclusions about the parameters that influence the flow behavior of NaCMC than has hitherto been available with mechanical methods. The flow birefringence, Δn ^′, rises as the degree of polymerization increases, and exhibits the same dependence on molar mass as does the viscosity: Δn ^′∝M _w ^3.4. As the degree of polymerization increases while the shear rate remains constant, the polymer segments become more distinctly aligned in the direction of shear. Hence increasing the degree of polymerization also affects the solution structure, i.e. the interaction of the molecules with one another. The stress-optical rule only applies to a limited extent for this system. The stress-optical coefficient, C, is almost independent of the shear rate, but is strongly influenced by the concentration and attains a limiting value of 3 × 10⁻⁸ Pa⁻¹. C was determined for a polymer in dilute solution and the curve obtained also enabled transitions in the solution structure to be recognized. Received: 1 May 1998 Accepted: 5 October 1998     

Perturbation for fractional-order evolution equation

Fractional calculus has gained a lot of importance during the last decades, mainly because it has become a powerful tool in modeling several complex phenomena from various areas of science and engineering. This paper gives a new kind of perturbation of the order of the fractional derivative with a study of the existence and uniqueness of the perturbed fractional-order evolution equation for ^CD^a-e₀₊u(t)=A ^CD^d₀₊u(t)+f(t),^{C}D^{\alpha-\epsilon}_{0+}u(t)=A~^{C}D^{\delta}_{0+}u(t)+f(t), u(0)=u _o , α∈(0,1), and 0≤ε, δ<α under the assumption that A is the generator of a bounded C _o -semigroup. The continuation of our solution in some different cases for α, ε and δ is discussed, as well as the importance of the obtained results is specified.     

ON THE CRITICAL POINTS OF THE MAP Fp:X→‖AXB-C‖pp

Suppose A,B and C are the bounded linear operators on a Hilbert space H, when A has a generalized inverse A^- such that (AA^-)^*=AA^- and B has a generalized inverse B^- such that (B^-B)^*=B^-B,the general characteristic forms for the critical points of the map F_p:X^→‖AXB-C‖_p^p(1p=2. Similarly, the same question has been discussed for several operators.     

Numerical solution of singular integral equations in stress concentration problems under longitudinal shear loading

Summary The paper deals with numerical solutions of singular integral equations in stress concentration problems for longitudinal shear loading. The body force method is used to formulate the problem as a system of singular integral equations with Cauchy-type singularities, where unknown functions are densities of body forces distributed in the longitudinal direction of an infinite body. First, four kinds of fundamental density functions are introduced to satisfy completely the boundary conditions for an elliptical boundary in the range 0≤φ _k ≤2π. To explain the idea of the fundamental densities, four kinds of equivalent auxiliary body force densities are defined in the range 0≤φ _k ≤π/2, and necessary conditions that the densities must satisfy are described. Then, four kinds of fundamental density functions are explained as sample functions to satisfy the necessary conditions. Next, the unknown functions of the body force densities are approximated by a linear combination of the fundamental density functions and weight functions, which are unknown. Calculations are carried out for several arrangements of elliptical holes. It is found that the present method yields rapidly converging numerical results. The body force densities and stress distributions along the boundaries are shown in figures to demonstrate the accuracy of the present solutions. Received 26 May 1998; accepted for publication 27 November 1998     

Differentiability Properties of Rotationally Invariant Functions

Let f be a function on the set Lin of all tensors (= square matrices) on a vector space of arbitrary dimension. If f is rotationally invariant (with respect to the left and right multiplication by proper orthogonal tensors), it has a representation through a symmetric even function of the signed singular values of the tensor argument A∈Lin. It is shown that f is of class C ^r ,r=0,1,...,∞, if and only if is of class C ^r , and an inductive formula is given for the derivatives D ^r f. This revised version was published online in August 2006 with corrections to the Cover Date.     

Experimental investigation of creep-crack-tip deformation using moiré interferometry

High-temperature moiré interferometry was applied to obtain full-field creep-crack-tip displacements of a three-point bend Al 2024-T4 specimen uner constant temperature of 200°C up to 720 hr.C ^* was evaluated by the moiré data obtained at discrete time intervals. Test results indicate that under steady-state creep condition, the creepcrack-tipv-displacement rate agrees with the asymptotic solution based on theC ^* integral. However, no creeping behavior was observed for the crack-tipu-displacement field aftert=276 hr. This discrepancy may be due to the initial large creep-crack-tip blunting and cavitation damage which alter the creep-crack-tip singular field such that theC ^* integral is no longer applicable to characterize the steady-state creep-crack-tip field. Postmortem studies of the tested specimen also revealed transgranular fracture path with the ordered grain boundaries perpendicular to the crack line, which may relate to the existence of the substantial crack-tip blunting and noncreeping behavior of theu-displacement field, i.e., the size and shape of material grain boundaries play an important role in the crack-tip-creeping behavior of the material.Paper was preseted at the 1991 SEM Spring Conference on Experimental Mechanics held in Milwaukee, WI on June 9–13.