On the collapse solution in limit analysis

The existence and regularity of collapse solutions in limit analysis of a plastic continuum is examined. Collapse fields for stresses and velocity exist as a saddle point for the duality between the static and kinematic formulations. The velocity field is defined as a pair u = (u , u ^T), where u is of bounded deformation in , while u^T is the velocity of the surface. A generalized Green's formula for the collapse fields is proved under certain regularity conditions.     

Collinear Central Configurations and Singular Surfaces in the Mass Space

For a given m=(m₁,...,m_n)(R⁺)ⁿ, let p and q(R³)ⁿ be two central configurations for m. Then we call p and q equivalent and write pq if they differ by an SO(3) rotation followed by a scalar multiplication as well as by a permutation of bodies. Denote by L(n,m) the set of equivalent classes of n-body collinear central configurations in R³ for any given mass vector m=(m₁,...,m_n)(R⁺)ⁿ. The main discovery in this paper is the existence of a union H₃ of three non-empty algebraic surfaces in the mass half space (m₁,m₂–m₁,m₃–m₂)R⁺×R² besides the planes generated by equal masses, which decreases the number of collinear central configurations. The union H₃ in R⁺×R ² is explicitly constructed by three 6-degree homogeneous polynomials in three variables such that, for any mass vector m=(m₁,m₂,m₃)(R⁺)³, ^# L(3,m)=3, if m₁, m₂, and m₃ are mutually distinct and (m₁,m₂–m₁,m₃–m₂)H₃, ^# L(3,m)=2, if m₁, m₂, and m₃ are mutually distinct and (m₁,m₂–m₁,m₃–m₂)H₃, ^# L(3,m)=2, if two of m₁, m₂, and m₃ are equal but not the third, ^# L(3,m)=1, if m₁=m₂=m₃. We give also a sharp upper bound on ^#L(n,m) for any positive mass vector m(R⁺)ⁿ.     

A Note on the Existence of Solutions to the Oseen System in Lipschitz Domains

We study the boundary-value problem associated with the Oseen system in the exterior of m Lipschitz domains of an euclidean point space We show, among other things, that there are two positive constants and α depending on the Lipschitz character of Ω such that: (i) if the boundary datum a belongs to L^q(∂Ω), with q ∈ [2,+∞), then there exists a solution (u, p), with and u ∈ L^∞(Ω) if a ∈ L^∞(∂Ω), expressed by a simple layer potential plus a linear combination of regular explicit functions; as a consequence, u tends nontangentially to a almost everywhere on ∂Ω; (ii) if a ∈ W^1-1/q,q(∂Ω), with then ∇u, p ∈ L^q(Ω) and if a ∈ C^0,μ(∂Ω), with μ ∈ [0, α), then also, natural estimates holds.     

Hopf Bifurcation in the presence of symmetry

Using group theoretic techniques, we obtain a generalization of the Hopf Bifurcation Theorem to differential equations with symmetry, analogous to a static bifurcation theorem of Cicogna. We discuss the stability of the bifurcating branches, and show how group theory can often simplify stability calculations. The general theory is illustrated by three detailed examples: O(2) acting on R ², O(n) on R ⁿ , and O(3) in any irreducible representation on spherical harmonics.The work of second author was also supported by a visiting position in the Department of Mathematics, University of Houston     

Asymptotic Stability of Riemann Solutions for a Class of Multidimensional Systems of Conservation Laws with Viscosity

We prove the asymptotic stability of two-state nonplanar Riemann solutions for a class of multidimensional hyperbolic systems of conservation laws when the initial data are perturbed and viscosity is added. The class considered here is those systems whose flux functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. In particular, we obtain the uniqueness of the self-similar L^∞ entropy solution of the two-state nonplanar Riemann problem. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in L_loc¹ of the space of directions ξ = x/t. That is, the solution u(t, x) of the perturbed problem satisfies u(t, tξ)→R(ξ) as t→∞, in L_loc¹(ℝⁿ), where R(ξ) is the self-similar entropy solution of the corresponding two-state nonplanar Riemann problem.     

Regularity of the Inverse of a Planar Sobolev Homeomorphism

Let be a domain. Suppose that f ∈ W^1,1_loc(Ω,R²) is a homeomorphism such that Df(x) vanishes almost everywhere in the zero set of J_f. We show that f^-1 ∈ W^1,1_loc(f(Ω),R²) and that Df⁻¹(y) vanishes almost everywhere in the zero set of Sharp conditions to quarantee that f⁻¹ ∈ W^1,q(f(Ω),R²) for some 1<q≤2 are also given.     

Dipole moments and near field potentials

Expressions are derived for the moments of electric and magnetic dipoles whose far field is equivalent to the Rayleigh term in the far field of a given radiating electromagnetic field. The equivalent dipole moments are expressed as integrals over an arbitrary closed surface of the near electromagnetic fields and simplification is achieved by rewriting these expressions in terms of the static potentials from which the near fields are derived. The results are valid regardless of the complexity of the field and the medium within the integration surface as long as the exterior is homogeneous, isotropic and source-free.Nomenclature B smooth closed bounded surface in 3-space - E, H radiating electric and magnetic field vectors - E ₀, H ₀ static near field limits of E and H - E ⁱ incident electric field vector - E ₀ ^t , H ₀ ^t total static field (incident plus scattered) - F, f arbitrary vector and scalar functions of position - k wave number - unit normal from B to exterior - p, m normalized electric and magnetic dipole moments - r position vector of field point with respect to origin in B - unit vector in direction of r - r magnitude of r - r _B position vector of point on B - V volume enclosed by B - , scalar potentials for E ₀ and H ₀ respectively - ^(e), ^(m) electric and magnetic Hertz vectors Research sponsored by the U.S. Air Force Office of Scientific Research under Grant No. AFOSR 69-1794.     

Derivatives of the Stretch, Rotation and Exponential Tensors in n-Dimensional Vector Spaces

We present a solution for the tensor equation TX + XT ^T = H, where T is a diagonalizable (in particular, symmetric) tensor, which is valid for any arbitrary underlying vector space dimension n. This solution is then used to derive compact expressions for the derivatives of the stretch and rotation tensors, which in turn are used to derive expressions for the material time derivatives of these tensors. Some existing expressions for n = 2 and n = 3 are shown to follow from the presented solution as special cases. An alternative methodology for finding the derivatives of diagonalizable tensor-valued functions that is based on differentiating the spectral decomposition is also discussed. Lastly, we also present a method for finding the derivatives of the exponential of an arbitrary tensor for arbitrary n.     

The construction of dispersion approximations to the solution of the vector convective diffusion equation

This paper is the sequel to an earlier paper in which we discover how to construct dispersion approximations to the solution of the vector convective diffusion equation. In this paper we make the construction concrete by providing useful formulas for the important elements of our theory. For definiteness we investigate cylinders of circular cross-section.Notation re() real part of - im() imaginary part of - int[] greatest integer less than or equal to - Z* adjoint of Z - adj Z adjugate of Z - col Z column vector of Z - det Z determinant of Z - Z generalized inverse of Z - Im Z image of Z - [,] generalized inner product - Ker Z kernel of Z - Ker Z* subspace perpendicular to Ker Z* - rank Z rank of Z - (Z) spectral radius of Z - tr Z trace of Z     

An Eigenvalue Criterion for Stability of a Steady Navier–Stokes Flow in {\mathbb{R}}^3

We study the resolvent equation associated with a linear operator L{\mathcal{L}} arising from the linearized equation for perturbations of a steady Navier–Stokes flow U^*{\mathbf{U^*}}. We derive estimates which, together with a stability criterion from [33], show that the stability of U^*{\mathbf{U^*}} (in the L²-norm) depends only on the position of the eigenvalues of L{\mathcal{L}}, regardless the presence of the essential spectrum.     

A Navier–Stokes Approximation of the 3D Euler Equation with the Zero Flux on the Boundary

Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T ₀ > 0, ν ₀ > 0 and a unique continuous family of strong solutions u ^ν (0 ≤ ν < ν ₀) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T ₀). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary conditions imposed on curl u ^ν and curl ² u ^ν .      

On analytical solutions of the equation utt − cu xx + au + bu3 = 0

In this paper a set of the analytical solutions of the equationu _tt−cu _xx +au+bu ³=0 are given.     

The viscous air flow pattern in the Stefan diffusion tube

In this paper, we consider the problem of the binary viscous diffusion of vapour through a Stefan tube, which is the model of an elementary capillary. While some preceding results in particular cases supposed parabolic velocity profiles and showed air recirculation, we treat here the general problem of a tube of finite length, submitted to a double viscous diffusion of vapour and air from a liquid surface. The movement of gas is expressed with conservation equations and ideal gas equations. The following added restrictions: constant temperature, no buyoancy effect, no inertial forces, are compatible with a capillary. A numerical solution based on the control volume method is obtained at every point in the tube. The results give the vapour and air flux, describe the circulation pattern and show that the vapour profile of concentration is level. In the lower part of the cylindrical tube space, over a distance of the length of a radius an important radial movement occurs, due to the recirculation of air which changes direction once it reaches the liquid surface. The velocity profile of the gas flow then becomes parabolic in the upper part of the tube.In order to easily obtain a numerical solution, the system of dimensionless equations is expanded to a series and transformed into a set of sub-systems. The little parameter used for this expansion is tied to the vapour concentration on the liquid surface. The solution of the sub-system of order zero, which is easier to compute, represents a good approximation of the complete solution. These solutions are situated in comparison with the Stefan diffusion and show that the influence of the viscous effect on the vapour flux is limited to a few percents.In order to apply the results to porous media where the pores are not so regular, we consider at last the diffusion in a tube including a contracted section in the middle of the tube. Since the diffusion paths are longer, the vapour flux is reduced, while the viscosity effect becomes more considerable. The reduction of the air flux is more significant than that of the vapour. This part of the study provides a better understanding of the diffusion through the pits at the wall fiber, and gives data for the air flux which permeates into the oak wood and produces tannin oxidation and thus discolouration.Nomenclature m _v/ vapour concentration - m _l vapour concentration in equilibrium with its liquid phase - D coefficient of molecular diffusion of a vapour in air (m²/s) - J _a vector density of mass flux of dry air (kg/m²s) - J _v vector density of mass flux of vapour (kg/m²s) - L Capillary length (m) - M _a dry air molar mass (kg/mole) - M _v Vapour molar mass (kg/mole) - P _atm atmospheric pressure - P gas mixture total pressure (Pascal) - R Ideal gas constant (J/mole K) - r _a= R/M _a,r _v=R/M _v - r ₀ tube or capillary radius (m) - T Temperature (K) - u axial component ofV - V gas mixture velocity vector (m/s) - v radial component ofV Greek Letters density of gas (kg/m³) - gas mixture dynamic viscosity (kg/ms) Numerical Values of Parameters D 3×10^–5m²/s (water vapour in air) - 2×10^–5kg/ms - M _a 29×10^–3kg/mole - M _v 18×10^–3kg/mole - T 323K - R 8.31 J/moleK - P _atm 10⁵Pa     

Two-phase geothermal flows with conduction and the connection with Buckley-Leverett theory

Two-phase flows of boiling water and steam in geothermal reservoirs satisfy a pair of conservation equations for mass and energy which can be combined to yield a hyperbolic wave equation for liquid saturation changes. Recent work has established that in the absence of conduction, the geothermal saturation equation is, under certain conditions, asymptotically identical with the Buckley-Leverett equation of oil recovery theory. Here we summarise this work and show that it may be extended to include conduction. In addition we show that the geothermal saturation wave speed is under all conditions formally identical with the Buckley-Leverett wave speed when the latter is written as the saturation derivative of a volumetric flow.Roman Letters C(P, S,q) geothermal saturation wave speed [ms^–1] (14) - c _t (P, S) two-phase compressibility [Pa^–1] (10) - D(P, S) diffusivity [m s^–2] (8) - E(P, S) energy density accumulation [J m^–3] (3) - g gravitational acceleration (positive downwards) [ms^–2] - h _w (P),h _w (P) specific enthalpies [J kg^–1] - J _M (P, S,P) mass flow [kg m^–2 s^–1] (5) - J _E (P, S,P) energy flow [J m^–2s^–1] (5) - k absolute permeability (constant) [m²] - k _w (S),k _s (S) relative permeabilities of liquid and vapour phases - K formation thermal conductivity (constant) [Wm^–1 K^–1] - L lower sheetC<0 in flow plane - m, c gradient and intercept - M(P, S) mass density accumulation [kg m^–3] (3) - O flow plane origin - P(x,t) pressure (primary dependent variable) [Pa] - q volume flow [ms^–1] (6) - S(x, t) liquid saturation (primary dependent variable) - S ^*(x,t) normalised saturation (Appendix) - t time (primary independent variable) [s] - T temperature (degrees Kelvin) [K] - T _sat(P) saturation line temperature [K] - TdT _sat/dP saturation line temperature derivative [K Pa^–1] (4) - T _c ,T _D convective and diffusive time constants [s] - u _w (P),u _s (P),u _r (P) specific internal energies [J kg^–1] - U upper sheetC > 0 in flow plane - U(x,t) shock velocity [m s^–1] - x spatial position (primary independent variable) [m] - X representative length - x, y flow plane coordinates - z depth variable (+z vertically downwards) [m] Greek Letters _P , _S remainder terms [Pa s^–1], [s^–1] - double-valued saturation region in the flow plane - h =h _s –h _w latent heat [J kg^–1] - = _w – _s density difference [kg m^–3] - line envelope - =D _K /D ₀ diffusivity ratio - porosity (constant) - _w (P), _s (P), _t (P, S) dynamic viscosities [Pa s] - v _w (P),v _s (P) kinematic viscosities [m²s^–1] - v ₀ =kh/KT kinematic viscosity constant [m² s^–1] - ₀ =v ₀ dynamic viscosity constant [m² s^–1] - _w (P), _s (P) density [kg m^–3] Suffixes r rock matrix - s steam (vapour) - w water (liquid) - t total - av average - 0 without conduction - K with conduction     

Stability estimates of the Boltzmann equation with quantum effects

We study uniform stability estimates to the Boltzmann equation for quantum particles such as Bose-Einstein particles and Fermi-Dirac particles. When the small amount of particles expands toward the vacuum, we show that continuous mild solutions are L ¹-stable and also satisfy BV-type estimates using a nonlinear functional approach. PACS05.20 Dd     

Some bounded results of θ(t)-type singular integral operators

Let B be a Banach space in UMD with an unconditional basis. The boundedness of the θ(t)-type singular integral operators in L _B ^p (R ⁿ), (1≤p<+∞) and H _B ¹ (R ⁿ) spaces are discussed. Foundation item: the Education Commission of Shandong Province (J98P51) Biography: Zhao Kai (1960-)     

Homeomorphisms of Bounded Variation

We show that the inverse of a planar homeomorphism of bounded variation is also of bounded variation. In higher dimensions we show that f ⁻¹ is of bounded variation provided that f ϵ W ^1,1(Ω; R ⁿ ) is a homeomorphism with |Df| in the Lorentz space L ^n-1,1(Ω). Dedicated to Tadeusz Iwaniec on his 60th birthday     

On K-BKZ and other viscoelastic models as continuum generalizations of the classical spring-dashpot models

An alternate constitutive formulation for visco-elastic materials, with particular emphasis on macromolecular viscoelastic fluids, is presented by generalizing Maxwell's idealized separation of elastic and relaxation mechanisms. The notion ofrelative rate of change of elastic stress is identified, abstracted, and formulated with the help of the established theory of finitely elastic isotropic materials. This given a local rate-type constitutive relation for an elastic mechanism in a simple material.For the simplest class of viscoelastic polymer melts, the notion of rate of change of elastic stress and its damped accumulation is identified and formulated. Under conditions of moderate strain rates, this scheme implies the reliable K-BKZ model for a class of polymer melts. An obvious extension generalizes the remaining classical spring-dashpot models. I Set of second-order tensors.A I is identified with a 3 × 3 matrix in a Cartesian co-ordinate system - I ^sym Set of symmetric second order tensors - Q Orthogonal tensor, i.e.Q ^T=Q ^–1. - Symbol for the value of the functional H:X I ^sym, whereX is the set of piecewise continuous and differentiable strain historiesF _to : [t ₀,t] I Other functionals, unless otherwise specified, should be interpreted in a similar manner.     

On Determinants having Complex Conjugate Columns or Rows

It is proved that the determinant, det D, of an N × N matrix D having n ≤ N/2 pairs of complex conjugate columns (or rows), while all other elements are real-valued, is given by det D =(−2i)ⁿdet S,n = 0,1,2,... in which S is a certain residual matrix having real-valued elements. Thus, det D is either real-valued or pure imaginary according as n is even (including n = 0 ) or odd, respectively. The general theorem is illustrated in an example. This revised version was published online in August 2006 with corrections to the Cover Date.