Nonlinear Kelvin-Helmholtz instability of two miscible ferrofluids in porous media

The nonlinear theory of the Kelvin-Helmholtz instability is employed to analyze the instability phenomenon of two ferrofluids through porous media. The effect of both magnetic field and mass and heat transfer is taken into account. The method of multiple scale expansion is employed in order to obtain a dispersion relation for the first-order problem and a Ginzburg–Landau equation, for the higher-order problem, describing the behavior of the system in a nonlinear approach. The stability criterion is expressed in terms of various competing parameters representing the mass and heat transfer, gravity, surface tension, fluid density, magnetic permeability, streaming, fluid thickness and Darcy coefficient. The stability of the system is discussed in both theoretically and computationally, and stability diagrams are drawn. Received: July 25, 2002; revised: April 16, 2003     

Density-driven compactional flow in porous media

In the mathematical modelling of compactional flow in porous media, the constitutive relation is typically modelled in terms of a nonlinear relationship between effective pressure and porosity, and compaction is essentially poroelastic. However, at depths deeper than 1 km where the pressure is high, compaction becomes more akin to a viscous one. Two mathematical models of compaction in porous media are formulated and the nonlinear equations are then solved numerically. The essential features of numerical profiles of poroelastic and viscous compaction are thus compared with asymptotic solutions. Two distinguished styles of density-driven compaction in fast and slow compacting sediments are analysed and shown in this paper.     

Stochastic partial differential equations in bounded domains with Dirichlet boundary conditions

In this paper we prove the existence of a unique solution for a class of stochastic parabolic partial differential equations in bounded domains, with Dirichlet boundary conditions. The main tool is an equivalence result, provided by the stochastic characteristics method, between the stochastic equations under investigation and a class of deterministic parabolic equations with moving boundaries, depending on random coefficients. We show the existence of the solution to this last problem, thus providing a solution to the former.     

Mixed boundary value problem of Laplace equation in a bounded Lipschitz domain

We study the existence and uniqueness of the mixed boundary value problem for Laplace equation in a bounded Lipschitz domain Ω⊂Rn, n?3. Let the boundary ∂Ω of Ω be decomposed by , Γ₁∩Γ₂=∅. We will show that if the Neumann data ψ is in and the Dirichlet data f is in , then the mixed boundary value problem has a unique solution and the solution is represented by potentials.     

Complete bounded trajectories and attractors for compressible barotropic self-gravitating fluid

In this paper, we consider the global behavior of weak solutions of the Navier-Stokes system of compressible barotropic self-gravitating fluids in time in a bounded three dimension domain-arbitrary forces. Under certain restrictions imposed on the adiabatic constant γ, we prove the existence of global compact attractors.     

Separations of first and second order theories in bounded arithmetic

We prove that PTC_N(n) (the polynomial time closure of the nonstandard natural number n in the model N of S₂.) cannot be a model of U¹₂. This implies that there exists a first order sentence of bounded arithmetic which is provable in U¹₂ but does not hold in PTC_N(n).     

Nonlinear systems arising from nonisothermal, non-Newtonian Hele-Shaw flows in the presence of body forces and sources

In this paper, we give a formal derivation of several systems of equations for injection moulding. This is done starting from the basic equations for nonisothermal, non-Newtonian flows in a three-dimensional domain. We derive systems for both (T⁰, p⁰) and (T¹, p¹) in the presence of body forces and sources. We find that body forces and sources have a nonlinear effect on the systems. We also derive a nonlinear “Darcy law”. Our formulation includes not only the pressure gradient, but also body forces and sources, which play the role of a nonlinearity. Later, we prove the existence of weak solutions to certain boundary value problems and initial-boundary value problems associated with the resulting equations for (T⁰,p⁰) but in a more general mathematical setting.     

Interior-Point Methods for Nonconvex Nonlinear Programming: Filter Methods and Merit Functions

Recently, Fletcher and Leyffer proposed using filter methods instead of a merit function to control steplengths in a sequential quadratic programming algorithm. In this paper, we analyze possible ways to implement a filter-based approach in an interior-point algorithm. Extensive numerical testing shows that such an approach is more efficient than using a merit function alone.     

Statistical Inference in Single-Index and Partially Nonlinear Models

A finite series approximation technique is introduced. We first applythis approximation technique to a semiparametric single-index model toconstruct a nonlinear least squares (LS) estimator for an unknown parameterand then discuss the confidence region for this parameter based on theasymptotic distribution of the nonlinear LS estimator. Meanwhile, acomputational algorithm and a small sample study for this nonlinear LSestimator are developed. Additionally, we apply the finite seriesapproximation technique to a partially nonlinear model and obtain some newresults.     

非光滑非线性互补问题的牛顿法

研究了非光滑的非线性互补问题. 首先将非光滑的非线性互补问题转化为一个非光滑方程组,然后用牛顿法求解这个非光滑方程组. 在该牛顿法中,每次迭代只需一个原始函数B-微分中的一个元素. 最后证明了该牛顿法的超线性收敛性.     

FILTRATION IN PARTIALLY SATURATED AND PARTIALLY DRIED LAYERED POROUS MEDIA

1.IntroductionThefiltrationprobleminInferedporousmediaarisesfromthestudiesofwatermovementduringirrigationandofthesalinizationofsoil.Thisproblemhasbeenelaboratelyinvestigatedforthesaturatedcase,whileforthegeneralcase,worksseemtoconcentrateonlyonexperimentalandnumericalaspects.Theseworksrevealsomeinterestingtheoreticalquestions.FOrexample,HillandParlange[2]foundin1972thattheverticalinfiltrationofwaterintwo-layeredsandconstitutedwithfinerupperlayerandcoarserlowerisunstable.Afterthemathematicali…     

Modelling boundary and nonlinear effects in porous media flow

We consider the problem of employing a Brinkman–Forchheimer system to model flow in a porous medium when Newton cooling conditions are appropriate at the boundary of the body. Specifically it is shown that the solution depends continuously on the Forchheimer coefficient and on the coefficient in the Newton cooling law at the boundary. Since we are dealing with non-slow flow rates and a porosity which is close to one we employ the Brinkman–Forchheimer equations and this leads to a second order differential inequality in the analysis as opposed to the first order one often found.     

Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications

A maximum principle is proved for the weak solutions of the telegraph equation in space dimension three u_tt−Δ_xu+cu_t+λu=f(t,x), when c>0, λ∈(0,c²/4] and (Theorem 1). The result is extended to a solution and a forcing belonging to a suitable space of bounded measures (Theorem 2). Those results provide a method of upper and lower solutions for the semilinear equation u_tt−Δ_xu+cu_t=F(t,x,u). Also, they can be employed in the study of almost periodic solutions of the forced sine-Gordon equation. A counterexample for the maximum principle in dimension four is given.     

Ordinal notations and well-orderings in bounded arithmetic

This paper investigates provability and non-provability of well-foundedness of ordinal notations in weak theories of bounded arithmetic. We define a notion of well-foundedness on bounded domains. We show that T₂¹ and S₂² can prove the well-foundedness on bounded domains of the ordinal notations below ₀ and Γ₀. As a corollary, the class of polynomial local search problems, PLS, can be augmented with cost functions that take ordinal values below ₀ and Γ₀ without increasing the class PLS.     

On the Kirchhoff plates equations with thermal effects and memory boundary conditions

In this paper we study the existence of weak and strong global solutions and uniform decay of the energy to the Kirchhoff plates equations with thermal effect and memory conditions working at the boundary. We show that the dissipation produced by the memory effect not depend on the present values of temperature gradient. That is, we show that the dissipation produced by memory effect is strong enough to produce exponential decay of the solution provided the relaxation functions also decays exponentially. When the relaxation functions decays polynomially, we show that the solution decays polynomially with the same rate.     

A multicomponent flow model in deformable porous media

We propose a model for multicomponent flow of immiscible fluids in a deformable porous medium accounting for capillary hysteresis. Oil, water, and air in the soil pores offer a typical example of a real situation occurring in practice. We state the problem within the formalism of continuum mechanics as a slow diffusion process in Lagrange coordinates. The balance laws for volumes, masses, and momentum lead to a degenerate parabolic PDE system. In the special case of a rigid solid matrix material and three fluid components, we prove under further technical assumptions that the system is mathematically well posed in a small neighborhood of an equilibrium.     

Robust Multiscale Iterative Solvers for Nonlinear Flows in Highly Heterogeneous Media

In this paper, we study robust iterative solvers for finite element systems resulting in approximation of steady-state Richards' equation in porous media with highly heterogeneous conductivity fields. It is known that in such cases the contrast, ratio between the highest and lowest values of the conductivity, can adversely affect the performance of the preconditioners and, consequently, a design of robust preconditioners is important for many practical applications. The proposed iterative solvers consist of two kinds of iterations, outer and inner iterations. Outer iterations are designed to handle nonlinearities by linearizing the equation around the previous solution state. As a result of the linearization, a large-scale linear system needs to be solved. This linear system is solved iteratively (called inner iterations), and since it can have large variations in the coefficients, a robust preconditioner is needed. First, we show that under some assumptions the number of outer iterations is independent of the contrast. Second, based on the recently developed iterative methods, we construct a class of preconditioners that yields convergence rate that is independent of the contrast. Thus, the proposed iterative solvers are optimal with respect to the large variation in the physical parameters. Since the same preconditioner can be reused in every outer iteration, this provides an additional computational savings in the overall solution process. Numerical tests are presented to confirm the theoretical results.     

Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws

We present a new approach to analyze the validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws whose eigenvalues are allowed to have constant multiplicity and corresponding characteristic fields to be linearly degenerate. The approach is based on our careful construction of more accurate auxiliary approximation to weakly nonlinear geometric optics, the properties of wave front-tracking approximate solutions, the behavior of solutions to the approximate asymptotic equations, and the standard semigroup estimates. To illustrate this approach more clearly, we focus first on the Cauchy problem for the hyperbolic systems with compact support initial data of small bounded variation and establish that the L ¹-estimate between the entropy solution and the geometric optics expansion function is bounded by O(?²), independent of the time variable. This implies that the simpler geometric optics expansion functions can be employed to study the behavior of general entropy solutions to hyperbolic systems of conservation laws. Finally, we extend the results to the case with non-compact support initial data of bounded variation.