Interface Vanishing for Solutions to Maxwell and Stokes Systems

This paper is concerned with the component-wise regularity of the solution to the stationary Maxwell or Stokes systems. We assume that there is a surface in R³, regarded as an interface, and the solution u to one of the systems is smooth except for this . Then, under these assumptions, we can show that some components of u are smooth across . In the Maxwell system, the normal component of u is always regular across . On the other hand, in the Stokes system, the singularity of u across can only arise to the normal derivatives of its tangential components. Furthermore, these results are shown to be optimal.     

Relations Between Energy and Enstrophy on the Global Attractor of the 2-D Navier-Stokes Equations

We examine how the global attractor of the 2-D periodic Navier–Stokes equations projects in the normalized, dimensionless energy–enstrophy plane (e, E). We treat time independent forces, with the view of understanding how the attractor depends on the nature of the force. First we show that for any force, is bounded by the parabola E = e^1/2 and the line E=e. We then show that for to have points near enough to the parabola, the force must be close to an eigenvector of the Stokes operator A; it can intersect the parabola only when the force is precisely such an eigenvector, and does so at a steady state parallel to this force. We construct a thin region along the parabola, pinched at such steady states, that the attractor can never enter. We show that 0 cannot be on the attractor unless the force is in H^m for all m. Different lower bound estimates on the energy and enstrophy on are derived for both smooth and nonsmooth forces, as are bounds on invariant sets away from 0 and near the line E = e. Motivation for the particular attention to the regions near the parabola and near 0 comes from turbulence theory, as explained in the introduction. Mathematics Subject Classification (2000): 35Q30, 76F02.     

Unexpected Behavior of Nonautonomous Linear Dynamical Systems in Two Dimensions

We show two examples of systems in with such that |Z_t| is strictly decreasing in time for any n but as .     

Superstable Manifolds of Semilinear Parabolic Problems

We investigate the dynamics of the semiflow φ induced on H₀¹(Ω) by the Cauchy problem of the semilinear parabolic equation

Dimensional analysis of pore scale and field scale immiscible displacement

A basic re-examination of the traditional dimensional analysis of microscopic and macroscopic multiphase flow equations in porous media is presented. We introduce a macroscopic capillary number which differs from the usual microscopic capillary number Ca in that it depends on length scale, type of porous medium and saturation history. The macroscopic capillary number is defined as the ratio between the macroscopic viscous pressure drop and the macroscopic capillary pressure. can be related to the microscopic capillary number Ca and the LeverettJ-function. Previous dimensional analyses contain a tacit assumption which amounts to setting = 1. This fact has impeded quantitative upscaling in the past. Our definition for , however, allows for the first time a consistent comparison between macroscopic flow experiments on different length scales. Illustrative sample calculations are presented which show that the breakpoint in capillary desaturation curves for different porous media appears to occur at 1. The length scale related difference between the macroscopic capillary number for core floods and reservoir floods provides a possible explanation for the systematic difference between residual oil saturations measured in field floods as compared to laboratory experiment.     

On the continuum modelling of porous media containing fluid: a molecular viewpoint with particular attention to scale

Mass conservation and linear momentum balance relations for a porous body and any fluid therein, valid at any given length scale in excess of nearest-neighbour molecular separations, are established in terms of local weighted averages of molecular quantities. The mass density field for the porous body at a given scale is used to identify its boundary at this scale, and a porosity field is defined for any pair of distinct length scales. Specific care is paid to the interpretation of the stress tensor associated with each of the body and fluid at macroscopic scales, and of the force per unit volume each exerts on the other. Consequences for the usual microscopic and macroscopic viewpoints are explored.Nomenclature material system; Section 2.1. - porous body (example of a material system); Sections 2.1, 3.1, 4.1 - fluid body (example of a material system); Sections 2.1, 3.1, 4.1 - weighting function; Sections 2.1, 2.3 - ,h weighting function corresponding to spherical averaging regions of radius and boundary mollifying layer of thicknessh; Section 3.2 - Euclidean space; Section 2.1 - V space of all displacements between pairs of points in; Section 2.1 - mass density field corresponding to; (2.3)₁ - ^P , ^f mass density fields for , ; (4.1) - P momentum density field corresponding to; (2.3)₂ - v velocity field corresponding to; (2.4) - S _r (X) interior of sphere of radiusr with centre at pointx; (3.3) - boundary ofany region - region in which ^p > 0 with = ,h; (3.1) - subset of whose points lie at least+h from boundary of ; (3.4) - abbreviated versions of ; Section 3.2, Remark 4 - strict interior of ; (3.7) - analogues of for fluid system ; Section 3.2 - general version of corresponding to any choice of weighting function; (4.6) - interfacial region at scale; (3.8) - ₀ scale of nearest-neighbour separations in ; Section 3.2. Remark 1 - porosity field at scales ( ₁; ₂); (3.9) - pore space at scales ( ₁; ₂); (3.12)     

Existence of Weak Solutions to the Equations of Non-Stationary Motion of Non-Newtonian Fluids with Shear Rate Dependent Viscosity

In the present paper we prove the existence of weak solutions to the equations of non-stationary motion of an incompressible fluid with shear rate dependent viscosity in a cylinder Q = Ω × (0,T), where denotes an open set. For the power-low model with we are able to construct a weak solution with ∇ · u = 0.     

On the Local Smoothness of Solutions of the Navier–Stokes Equations

We consider the Cauchy problem for incompressible Navier–Stokes equations with initial data in , and study in some detail the smoothing effect of the equation. We prove that for T < ∞ and for any positive integers n and m we have , as long as stays finite.     

Strong Solutions for Generalized Newtonian Fluids

We consider the motion of a generalized Newtonian fluid, where the extra stress tensor is induced by a potential with p-structure (p = 2 corresponds to the Newtonian case). We focus on the three dimensional case with periodic boundary conditions and extend the existence result for strong solutions for small times from \tfrac{5}{3}$$ " align="middle" border="0"> (see [16]) to \tfrac{7}{5}.$$ " align="middle" border="0"> Moreover, for we improve the regularity of the velocity field and show that for all 0.$$ " align="middle" border="0"> Within this class of regularity, we prove uniqueness for all \tfrac{7}{5}.$$ " align="middle" border="0"> We generalize these results to the case when p is space and time dependent and to the system governing the flow of electrorheological fluids as long as      

Traveling Waves of a Singularly Perturbed System of Integral–Differential Equations Arising from Neuronal Networks

A nonlinear nonlocal model arising from synaptically coupled neuronal networks with two integral terms is considered. The existence and stability of several traveling wave solutions are established by using ideas in differential equations and functional analysis. Steady-state solutions of some inhomogeneous integral–differential equations are also investigated. We consider several types of kernel functions: (I) positive functions, such as and , where ρ>0 is a constant; (II) nonnegative kernels with compact supports, for examples, (i) 1$$" align="middle" border="0"> , and (ii) {\pi\over 2}$$" align="middle" border="0"> ; (III) Mexican hat type kernel functions, such as and , where A>B>0 and a>b>0 are constants.Dedicated to Professor Yulin Zhou and Professor Boling Guo on the Occassions of their birthdays.     

A Numerical Study of the Applicability of the Shear Compression Specimen to Parabolic Hardening Materials

This paper addresses quasi-static loading of the shear compression specimen (SCS), that has been especially developed to investigate the shear response of materials at various strain rates. Previous work [4, 5] addressed bi-linear hardening materials, whereas the present work concerns parabolic hardening materials. The investigation is done numerically using three-dimensional elastoplastic finite element simulations. The analyses show that the averaged von Mises stress ( ) and strain ( ) on a mid-section of the gauge reflect accurately the prescribed parabolic hardening model. A method for finding the parabolic hardening coefficients and reducing the measured load, P, and displacement, d, into equivalent stress and strain is introduced and tested. A very good agreement is observed, thus confirming the potential of the technique for large strain testing of parabolic hardening materials.     

The Experimental Approach to Effective Stress Law of Coal Mass by Effect of Methane

Effective stress law of all kinds of coal samples, including steam coal, fat coal, corking coal, thin coal and anthracite, under pore pressure of gas, is experimentally studied using a newly developed test machine. These samples are taken from Coal Mines in Wuda, Hebi, Yanzhou, Yangquan, Qingshui, and Gujiao in China. The experiment results show that, under pore pressure of gas, the tested coal samples comply with Biots effective stress law, where the Biots coefficient is not a constant, and is bilinear function of volumetric stress () and pore pressure (p), that is, We define four areas according to the numerical feature of , that is, functionless area of pore pressure, normal function area, fracturing function area, and quasi-soil function area. The effective stress law of coal mass introduced by this paper is a constitutive equation in the study of coupled solid and fluid. This has significance in the drainage and outburst of methane in coal seam.     

The Connection Between Isometries and Symmetries of Geodesic Equations of the Underlying Spaces

A connection between the symmetries of manifolds and differential equations is sought through the geodesic equations of maximally symmetric spaces, which have zero, constant positive or constant negative curvature. It is proved that for a space admitting so(n+1) or so(n,1) as the maximal isometry algebra, the symmetry of the geodesic equations of the space is given by so( or (where d ₂ is the two-dimensional dilation algebra), while for those admitting (where represents semidirect sum) the algebra is sl(n+2). A corresponding result holds on replacing so(n) by so(p,q) with p+q = n. It is conjectured that if the isometry algebra of any underlying space of non-zero curvature is h, then the Lie symmetry algebra of the geodesic equations is given by , provided that there is no cross-section of zero curvature at the point under consideration. If there is a flat subspace of dimension m, then the symmetry group becomes ).     

Stokes and Navier–Stokes Equations with Robin Boundary Conditions in a Half-Space

We study the initial-boundary value problem for the Stokes equations with Robin boundary conditions in the half-space It is proved that the associated Stokes operator is sectorial and admits a bounded H^∞-calculus on As an application we prove also a local existence result for the nonlinear initial value problem of the Navier–Stokes equations with Robin boundary conditions.     

Time-Averaged Photoelastic Stress Analysis of the Ultrasonic Wave in a Strip

In this paper, a digital photoelastic system was employed to observe the stress distribution generated by an ultrasonic wave impinged at the edge of a strip. According to the classical photoelastic theory, gray level distribution of the photoelastic fringe patterns was formulated and expressed in the form of , where J₀ is the zeroth-order Bessel function. This new technique is called the time-averaged photoelastic method. To verify the proposed method, the distribution of was superimposed onto the distribution of gray level of the experimentally obtained photoelastic fringe pattern caused by standing wave only. Except regions near the center of fringes, well-matched results were found.     

Acoustics of Rotating Deformable Saturated Porous Media

We investigate wave propagation in elastic porous media which are saturated by incompressible viscous Newtonian fluids when the porous media are in rotation with respect to a Galilean frame. The model is obtained by upscaling the flow at the pore scale. We use the method of multiple scale expansions which gives rigorously the macroscopic behaviour without any prerequisite on the form of the macroscopic equations. For Kibel numbers A A(1), the acoustic filtration law resembles a Darcys law, but with a conductivity which depends on the wave frequency and on the angular velocity. The bulk momentum balance shows new inertial terms which account for the convective and Coriolis accelerations. Three dispersive waves are pointed out. An investigation in the inertial flow regime shows that the two pseudo-dilatational waves have a cut-off frequency.     

Periodic Trajectories near Degenerate Equilibria in the Hénon-Heiles and Yang-Mills Hamiltonian Systems

This paper deals with connected branches of nonstationary periodic trajectories of Hamilton equations

Density-Dependent Incompressible Fluids in Bounded Domains

This paper is devoted to the study of the initial value problem for density dependent incompressible viscous fluids in a bounded domain of with boundary. Homogeneous Dirichlet boundary conditions are prescribed on the velocity. Initial data are almost critical in term of regularity: the initial density is in W^1,q for some q > N, and the initial velocity has fractional derivatives in L^r for some r > N and arbitrarily small. Assuming in addition that the initial density is bounded away from 0, we prove existence and uniqueness on a short time interval. This result is shown to be global in dimension N = 2 regardless of the size of the data, or in dimension N ≥ 3 if the initial velocity is small. Similar qualitative results were obtained earlier in dimension N = 2, 3 by O. Ladyzhenskaya and V. Solonnikov in [18] for initial densities in W^1,∞ and initial velocities in with q > N.     

Two-Phase Inertial Flow in Homogeneous Porous Media: A Theoretical Derivation of a Macroscopic Model

The purpose of this article is to derive a macroscopic model for a certain class of inertial two-phase, incompressible, Newtonian fluid flow through homogenous porous media. Starting from the continuity and Navier–Stokes equations in each phase β and γ, the method of volume averaging is employed subjected to constraints that are explicitly provided to obtain the macroscopic mass and momentum balance equations. These constraints are on the length- and time-scales, as well as, on some quantities involving capillary, Weber and Reynolds numbers that define the class of two-phase flow under consideration. The resulting macroscopic momentum equation relates the phase-averaged pressure gradient to the filtration or Darcy velocity in a coupled nonlinear form explicitly given by