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-)     

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.     

Stability of Discrete Stokes Operators in Fractional Sobolev Spaces

Using a general approximation setting having the generic properties of finite-elements, we prove uniform boundedness and stability estimates on the discrete Stokes operator in Sobolev spaces with fractional exponents. As an application, we construct approximations for the time-dependent Stokes equations with a source term in L ^p (0, T; L ^q (Ω)) and prove uniform estimates on the time derivative and discrete Laplacian of the discrete velocity that are similar to those in Sohr and von Wahl [20]. On long leave from LIMSI (CNRS-UPR 3251), BP 133, 91403, Orsay, France.     

Lattice gas simulation of Darcy flow with memory-free collisions

A lattice gas algorithm is proposed for the simulation of water flow in the unsaturated zone. Microscopic dynamics of a two-dimensional model system are defined. Up to four fluid particles occupy the sites of a square lattice. At each time step, the particles are sent to neighbouring sites according to probabilistic rules which depend on the permeability and the potential but not on the input velocities of the particles. On the macroscopic scale, the flow is described by a diffusion term and a Darcy term. Several extensions including higher dimension are discussed.List of Symbols c ⁽ⁿ⁾ constant in the definition of the rejection probabilityP forn = 1,2,3 particles at a site 0 c ⁽ⁿ⁾ 1 - D diffusion constant - D vertical extent of the system, measured in cells - E _i vector connecting a site to its neighbour in directioni - i direction of a nearest neighbour site,i = 1,..., 4 - j direction of a nearest neighbour site,j = 1,..., 4 - j mass transport (fluid flow),j = v - j ^x x-component of the flowj - k(x) spatial dependence of the permeability, user defined under the constraint 0 k 1 - k () the part of the permeability which depends on the degree of saturation (seek) - k ⁽ⁿ⁾ (x) effective permeability at a sitex that holdsn particles - L horizontal extent of the system, measured in cells - l _mac macroscopic length scale, e.g. one meter - l _mic microscopic length scale (one lattice constant) - m integer number of time steps - n (x) number of particles at the lattice sitex - N _A total number of particles on all A-sites - P probability for rejection of a randomly selected direction or set of directions - p arithmetic mean of the probability for a site to receive a particle from a particular neighbour (the average is taken over the four neighbours) - p _i ⁽ⁿ⁾ probability that one out ofn particles at a site is sent in directioni - p _ij ⁽²⁾ probability that the two particles at a site are sent in directionsi andj - t time - t _mac macroscopic time scale, e.g. one day - t _mic microscopic time scale (one time step) - v fluid velocity - x space vector, mostly two-dimensional:x = (x, y) - x horizontal component ofx - y vertical component ofx - quotient of microscopic and macroscopic time scales,t _mic /t _mac - quotient of microscopic and macroscopic length scales,l _mic /l _mac - _i p + _i is the probability that a particle is received from the neighbour atx +E _i - K(X, ) effective permeability,k =k(x)k () - correlation length - degree of saturation, used synonymously with density (homogeneous porosity) - ₀ value of a homogeneous particle density - ø(x) external potential (user defined), ø = ^gr + ^mat - ø(x) arithmetic mean of the external potential at the four sites surroundingx - ø _i external potential at the sitex +E _i - total potential, = ø + ^den - ^gr(x) gravitational potential - ^mat(x) matrix potential - ^den() density-dependent potential - _n potential depending on the occupation number - ⁽ⁿ⁾ (x) probability that sitex is occupied byn particles - ₀ ^{(n) (n)} in a system with homogeneous particle density - mac macroscopic - mic microscopic     

Energy Flow in Formally Gradient Partial Differential Equations on Unbounded Domains

As an example of an extended, formally gradient dynamical system, we consider the damped hyperbolic equation u _tt+u _t=u+F(x, u) in R ^N , where F is a locally Lipschitz nonlinearity. Using local energy estimates, we study the semiflow defined by this equation in the uniformly local energy space H¹ _ul(R ^N )×L² _ul(R ^N ). If N2, we show in particular that there exist no periodic orbits, except for equilibria, and we give a lower bound on the time needed for a bounded trajectory to return in a small neighborhood of the initial point. We also prove that any nonequilibrium point has a neighborhood which is never visited on average by the trajectories of the system, and we conclude that any bounded trajectory converges on average to the set of equilibria. Some counter-examples are constructed, which show that these results cannot be extended to higher space dimensions.     

Towards a Unified Field Theory for Classical Electrodynamics

In this paper we introduce a model which describes the relation of matter and the electromagnetic field from a unitarian standpoint in the spirit of ideas of Born and Infeld. In this model, based on a semilinear perturbation of Maxwell equations, the particles are finite-energy solitary waves due to the presence of the nonlinearity. In this respect the matter and the electromagnetic field have the same nature. Finite energy means that particles have finite mass and this makes electrodynamics consistent with the special relativity. We analyze the invariants of the motion of the semilinear Maxwell equations (SME) and their static solutions. In the magnetostatic case (i.e., when the electric field E = 0 and the magnetic field H does not depend on time) SME are reduced to the semilinear equation where × denotes the curloperator, f is the gradient of a strictly convex smooth function f:R³R and A:R³R³ is the gauge potential related to the magnetic field H (H = × A). Due to the presence of the curl operator, (1) is a strongly degenerate elliptic equation. Moreover, physical considerations impel f to be flat at zero (f(0)=0) and this fact leads us to study the problem in a functional setting related to the Orlicz space L^p+L^q. The existence of a nontrivial finite- energy solution of (1) is proved under suitable growth conditions on f. The proof is carried out by using a suitable variational framework related to the Hodge splitting of the vector field A.We thank Marino Badiale and Charles Stuart for their useful suggestions.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> Stability of the Boltzmann Equation for the Hard-Sphere Model

We study the L¹ stability of classical solutions to the Boltzmann equation for a hard-sphere model, when initial datum is a small perturbation of a vacuum, and tends to zero exponentially fast at infinity in the phase space. For this, we introduce nonlinear functionals measuring potential interactions between particles with different velocities and L¹ distance between classical solutions. We use pointwise estimates for a solution and the gain term of a collision operator to control the time-evolution of nonlinear functionals.Dedicated to Marshall Slemrod on the occasion of his 60th birthday     

On Stability and Strong Convergence for the Spatially Homogeneous Boltzmann Equation for Fermi-Dirac Particles

 The paper considers the stability and strong convergence to equilibrium of solutions to the spatially homogeneous Boltzmann equation for Fermi-Dirac particles. Under a cutoff condition on the collision kernel, we prove a strong stability in L ¹ topology at any finite time interval, and, for hard and Maxwellian potentials, we prove that the solutions converge strongly in L ¹ to equilibrium under a high temperature condition. The basic tools used are moment-production estimates and the strong compactness of the collision gain term. (Accepted 25, October 2002) Published online March 14, 2003 Communicated by P.-L. Lions     

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.     

L 2-Stability Theory of the Boltzmann Equation near a Global Maxwellian

We present three a priori L ²-stability estimates for classical solutions to the Boltzmann equation with a cut-off inverse power law potential, when initial datum is a perturbation of a global Maxwellian. We show that L ²-stability estimates of classical solutions depend on Strichartz type estimates of perturbations and the non-positive definiteness of the linearized collision operator. Several well known classical solutions to the Boltzmann equation fit our L ²-stability framework.     

Stability of Viscous Profiles: Proofs Via Dichotomies

In this paper we give a self-contained approach to a nonlinear stability result, as t → ∞, for a viscous profile corresponding to a strong shock of a system of conservation laws. The initial perturbation is assumed to be small and to have zero mass. As t→ ∞, the solution with perturbed initial data is shown to approach the viscous profile in maximum norm.A complete proof of the stability result is given under slightly weaker assumptions than those in [Comm. Pure Appl. Math. LI (1998) 1397]; our assumptions, techniques, and results also differ from those in [Indiana Univ. Math. J. 47 (1998) 741]. To derive resolvent estimates for a linearized problem, we use the theory of exponential dichotomies for ODEs extensively. A main tool provided by this theory is a quantitative L ₁ perturbation theorem for dichotomies, which yields the delicate resolvent estimates for s near zero.When showing that the resolvent estimates imply nonlinear stability, we essentially follow the arguments in [Comm. Pure Appl. Math. LI (1998) 1397; SIAM J. Math. Anal. 20 (1999) 401], but note some simplifications.     

Kinetics of a Model Weakly Ionized Plasma in the Presence of Multiple Equilibria

We study, globally in time, the velocity distribution f(v,t) of a spatially homogeneous system that models a system of electrons in a weakly ionized plasma, subjected to a constant external electric field E. The density f satisfies a Boltzmann-type kinetic equation containing a fully nonlinear electron‐electron collision term as well as linear terms representing collisions with reservoir particles having a specified Maxwellian distribution. We show that when the constant in front of the nonlinear collision kernel, thought of as a scaling parameter, is sufficiently strong, then the L ¹ distance between f and a certain time-dependent Maxwellian stays small uniformly in t. Moreover, the mean and variance of this time‐dependent Maxwellian satisfy a coupled set of nonlinear ordinary differential equations that constitute the “hydrodynamical” equations for this kinetic system. This remains true even when these ordinary differential equations have non‐unique equilibria, thus proving the existence of multiple stable stationary solutions for the full kinetic model. Our approach relies on scale‐independent estimates for the kinetic equation, and entropy production estimates. The novel aspects of this approach may be useful in other problems concerning the relation between the kinetic and hydrodynamic scales globally in time. (Accepted September 3, 1996)     

Couple on a rotary oscillating sphere in a dusty gas

Rotary oscillations of a sphere about its vertical diameter in an infinite expanse of viscous, incompressible fluid with imbedded identical spherical particles are studied. The problem is solved by the method of separation of variables and explicit expressions for the velocity of fluid particles and the couple experienced by the sphere due to fluid stresses are obtained. The couple is expressed in terms of two parameters and graphs have been drawn to represent the variation in these parameters and some interesting conclusions are made.Nomenclature u ⁵ velocity of fluid particles - v ⁶ velocity of dust particles - P ⁵ pressure - ⁶ fluid density - t ⁹ time - m mass of a dust particle - N number density of dust particles - k ⁵ Stokes resistance coefficient - ³ viscosity - ⁶ kinematic viscosity - ⁷ particle relaxation time - ⁴ frequency of oscillation of the sphere - f ⁹ mass concentration of the dust particles     

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 .      

Simulating shear behavior of a sandy soil under different soil conditions

Understanding of soil shear behavior is very important in the field of agricultural machinery and soil dynamics. In this study, a discrete element model was developed using a simulation tool, Particle Flow Code in Three Dimensions (PFC^3D). The model simulates direct shear tests of soil and predicts soil shear behavior, in terms of shear forces and displacements. To determine and calibrate model parameters (stiffness of particles, strength and stiffness of bond between particles), laboratory direct shear tests were conducted to examine effects of soil moisture content and bulk density on shear behaviors of a sandy soil. Three soil moisture levels (0.02%, 13.0%, and 21.5%) and four bulk density levels (0.99, 1.28, 1.36, and 1.50 Mg/m³) were used in the tests. The test results showed that in general drier and denser soil conditions produced higher shear forces. Based on the test results, the bond strengths of the model particles were determined from soil cohesion and internal friction angle. The model particle stiffness was calibrated based on the yield forces from the tests. The calibrated particle stiffness varied from 1.0 × 10³ to 8.2 × 10³ N/m, depending on soil moisture and density levels. The bond stiffness calibrated was 1.0 × 10⁷ Pa/m for all soil conditions.     

Phase interface conditions in a massless electron-positron gas

The steady motion of a massless electron-positron gas in a self-consistent electromagnetic field is considered. In such a gas, two phases are formed for each of the components: a dynamic phase, in which the particles move with the speed of light, and a static phase, in which the particles move with a subluminal speed. The static phase can exist only within the capture region, in which the conditions (E,H)=0 andE ²<H ² are satisfied. Strong and weak discontinuities of the electromagnetic field on the boundary of the capture region (at the phase interface) are studied.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 165–172, January–February, 1990.     

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.     

On the Domain of Validity of Brinkman’s Equation

An increasing number of articles are adopting Brinkman’s equation in place of Darcy’s law for describing flow in porous media. That poses the question of the respective domains of validity of both laws, as well as the question of the value of the effective viscosity μ ^e which is present in Brinkman’s equation. These two topics are addressed in this article, mainly by a priori estimates and by recalling existing analyses. Three main classes of porous media can be distinguished: “classical” porous media with a connected solid structure where the pore surface S _p is a function of the characteristic pore size l _p (such as for cylindrical pores), swarms of low concentration fixed particles where the pore surface is a function of the characteristic particle size l _s , and fiber-made porous media at low solid concentration where the pore surface is a function of the fiber diameter. If Brinkman’s 3D flow equation is valid to describe the flow of a Newtonian fluid through a swarm of fixed particles or fibrous media at low concentration under very precise conditions (Lévy 1983), then we show that it cannot apply to the flow of such a fluid through classical porous media.     

Optimal Time Decay of the Vlasov–Poisson–Boltzmann System in $${\mathbb R^3}$$

The Vlasov–Poisson–Boltzmann System governs the time evolution of the distribution function for dilute charged particles in the presence of a self-consistent electric potential force through the Poisson equation. In this paper, we are concerned with the rate of convergence of solutions to equilibrium for this system over \mathbb R³{\mathbb R^3}. It is shown that the electric field, which is indeed responsible for the lowest-order part in the energy space, reduces the speed of convergence, hence the dispersion of this system over the full space is slower than that of the Boltzmann equation without forces; the exact L ²-rate for the former is (1 + t)^−1/4 while it is (1 + t)^−3/4 for the latter. For the proof, in the linearized case with a given non-homogeneous source, Fourier analysis is employed to obtain time-decay properties of the solution operator. In the nonlinear case, the combination of the linearized results and the nonlinear energy estimates with the help of the proper Lyapunov-type inequalities leads to the optimal time-decay rate of perturbed solutions under some conditions on initial data.     

Accurate estimate of disturbance amplitude variation from solution of minimal composite stability theory

Minimal composite theory (Proc. R. Soc. Lond., Ser. A, 453 2537–2549, 1979) shows that, at the lowest order in the reciprocal of the local flow Reynolds number R, the stability of a spatially developing similarity flow may be described by an ordinary differential equation in the similarity coordinate. It is, in principle, not possible to determine the dependence of the disturbance amplitude on the streamwise coordinate solely from such an ordinary differential equation. However, noting that, to O(R^−2/3), the dependence of the eigenfunction on the normal coordinate is identical in both the full non-parallel and minimal composite theories, and using a method due to Gaster, we show how the streamwise variation of disturbance amplitude can be determined to O(R⁻¹ without solving a partial differential equation, although knowledge of the partial differential operator is required. Comparison with the DNS results of Fasel and Konzelmann shows excellent agreement with the present results. Furthermore, especially in strong adverse pressure gradients, the present amplitude ratio estimates are within 3% of the full non-parallel theory, whereas the Orr–Sommerfeld results show an underestimate by 26%.This revised version was published online in May 2005. In the previous version, the published online date was missing. Moreover, the preliminary article pagination was deleted.