An Effective Model of a Layered Medium in Which Alternating Porous and Elastic Layers Are in Slide Contact

For a medium in which porous and elastic layers alternate and there is slide contact on the interfaces, an effective model is established. This model is of three phases and includes two elastic phases and one fluid phase. Specific features of this effective model are that two waves with triangular front sets propagate and the second (slow) longitudinal wave is absent in it. In the special case where the thickness of elastic layers is very small but they continue to work as barriers for fluid particles from porous layers, the effective model is of two phases, and one of the triangular front sets disappears. Bibliography: 14 titles.     

An Effective Model of a Fractured Medium

A homogeneous isotropic elastic medium intersected by three systems of fractures on which the jumps of stresses are proportional to displacements is considered. An effective model of this medium is described by equations differing from the respective equations of the elastic medium by additional terms. On the basis of the equations of the effective model, the wave field excited by a point source is established. An investigation of the integral representation of the wave field shows that the velocities of the longitudinal and transversal waves and of the Rayleigh wave are functions of the frequency and the wave numbers. Formulas for the phase and group velocities of these waves are derived. Bibliography: 3 titles.     

具强退化性的一般渗流方程分界面的存在性

The aim of this paper is to discuss the Cauchy problem for quasilinear degenerate parabloic equations of the form ut-△φ（u） N↑∑i=1aψi(u)/axi=0,where φ∈ C^1(R^1) is a strictly monotonically increasing function.Clearly,the above equation has strong degeneracy,i.e.,the set of zero points of φ′(.)is permitted to have zero measure.In particular,the existence of interfaces of solutions is obtained.     

Regularity of the Interfaces with Sign Changes of Solutions of the One-Dimensional Porous Medium Equation

In previous papers we considered the Cauchy problem for the one-dimensional evolution p-Laplacian equation for nonzero, bounded, and nonnegative initial data having compact support, and showed that after a finite time the set of spatial critical points of the nonnegative solution u=u(x, t) in {u>0} consists of one point, the spatial maximum point of u, and the curve of the spatial maximum points is continuous with respect to the time variable. Since the spatial derivative ∂_xu satisfies the porous medium equation with sign changes, the curve of the spatial maximum points is regarded as an interface with sign changes of ∂_xu. On the other hand, in a paper by M. Bertsch and D. Hilhorst (1991, Appl. Anal.41, 111-130) the interfaces where the solutions change their sign were studied in detail for the initial-boundary value problems of the generalized porous medium equation over two-dimensional cylinders. But the monotonicity of the initial data is assumed there. As is noted in Section 4 of our earlier work (1996, J. Math. Anal. Appl.203, 78-103), the monotonicity of ∂_xu(?, t) in some neighborhood of the spatial maximum point of u(?, t) cannot be assumed, and therefore, if this monotonicity for some large t>0 is proved, then by the method of Bertsch and Hilhorst (cited above) one may get more precise regularity properties of the curve of the spatial maximum points. The purpose of the present paper is twofold. One is to remove some monotonicity assumption for initial data in Bertsch and Hilhorst's theorem concerning the regularity of the interfaces with sign changes of solutions of the one-dimensional generalized porous medium equation. By comparing the solution with appropriate symmetric nonnegative solutions we shall get the monotonicity of the solution near the interface after a finite time. The other is as a by-product of the method to get C¹ regularity of the curves of the spatial maximum points of nonnegative solutions of the Cauchy problem for the evolution p-Laplacian equation for sufficiently large t.     

On Effective Models of Porous Stratified Media with a Sliding Contact Between Layers

In order to study wave propagation in porous layered media with a sliding contact between the elastic phases on the interfaces, effective models of these media are investigated. For these models, the front sets of four waves excited by point sources are established and formulas for the wave velocities along the axes are derived. The methods of constructing the front sets applied in this paper allow one to point out special features of these front sets such as loops and juts. The particular case where all of the layers are identical and a sliding contact occurs between layers is also considered. Bibliography: 8 titles.     

多孔介质中的B rinkman-Forchheimer模型的收敛性

本文研究Brinkman-Forchheimer方程组的结构稳定性.首先我们得到一些有用的先验界,然后在此基础上推出解所满足的微分不等式,最后建立了解对Brinkman系数v的收敛性结果.     

Global Solvability,Pattern Formation and Stability to a Chemotaxis-haptotaxis Model with Porous Medium Diffusion

Acta Mathematica Sinica, English Series - In this paper, we deal with the following chemotaxis-haptotaxis system modeling cancer invasion with nonlinear diffusion $$\left\{ {\matrix{{{u_t} = \Delta...     

带接种免疫的网络传染病的有效度模型

传染病动力学模型是分析疾病传播的有效方法,能够解释疾病流行因素,并为疾控提供理论指导,而复杂网络的兴起为传染病建模提供了更好的工具,是最近几年的研究热点.在Lindquist等人提出的网络有效度模型的基础上加以推广,引入疫苗接种,提出了一个新的网络传染病模型,称之为"SV+SIS"模型.利用The next generation matrix方法计算了模型的传播阈值,并与各种SIS,SIR模型进行比较,结果发现疫苗接种能够有效降低基本再生数,这说明疫苗接种比因感染而获得免疫的防控效果更好.研究结果为传染病动力学的建模和分析提供了很好的参考.     

多孔介质平板通道传热模型的两种求解方法

基于Brinkman Darcy扩展模型和非局部热平衡模型,考虑液相和固相含有内热源的情况,建立了多孔介质平板通道传热的一般模型.分别采用直接法和间接法将液相与固相能量方程解耦,进而求得充分发展传热条件下的多孔介质温度场.与直接解耦法相比,间接解耦法可在原始边界条件下求解二阶微分方程,更加简单易行.通过对无量纲温度表达式系数以及温度分布的比较,验证了两种求解方法的等价性.在两种极限情形下,间接法所得温度分布解析解与现有文献结果相当吻合,这也在一定程度上证明了所建模型更具一般性.参数分析表明,液固两相温差随着Biot数或有效导热系数比的增大而减小,Nusselt数随着内热源比的增大而减小.     

The Cauchy Problem of the Porous Medium Equation with Absorption

In this paper we study the existence of solution for the following Cauchy problem {u_t = Δu^m - u^p u(x,0) = u_0(x) We show how the growth condition of initial trace is determined by the absorption.     

On the Boundary Condition at the Surface of a Porous Medium

A theoretical justification is given for an empirical boundary condition proposed by Beavers and Joseph [1]. The method consists of first using a statistical approach to extend Darcy's law to non homogeneous porous medium. The limiting case of a step function distribution of permeability and porosity is then examined by boundary layer techniques, and shown to give the required boundary condition. In an Appendix, the statistical approach is checked by using it to derive Einstein's law for the viscosity of dilute suspensions.     

Convective Porous Medium Systems with Nonlinear Forcing at the Boundary

This paper deals with the global positive solutions of convective porous medium systems with nonlinear forcing at the boundary. Necessary and sufficient conditions on the global existence of all positive solutions are obtained.     

具吸收项的多孔隙介质方程解的渐近性

本文研究了具吸收项的多孔隙介质方程解的渐近性质,部分地解答了P.L.Pelelier等人关于此问题的一个猜测。     

Computation of Travelling Combustion Waves in a Porous Medium

Numerical solutions for travelling combustion waves in a porous medium are sought. The algorithm of computation is based on a shooting method used in an existence proof. The numerical result suggests that there is a limit for the inlet gas velocity below which no travelling wave solution can be constructed.     

Blow-up of the Solution for a Class of Porous Medium Equation with Positive Initial Energy

This paper deals with a class of porous medium equation

ut = Δum + f(u)$ut\hspace{0.17em}=\hspace{0.17em}\text{Δ}{u}^m\hspace{0.17em}+\hspace{0.17em}f\left(u\right)$

with homogeneous Dirichlet boundary conditions. The blow-up criteria is established by using the method of energy under the suitable condition on the function f(u).     

Thermal Convection in a Darcy Porous Medium with Anisotropic Spatially Varying Permeability

We consider the problem of thermal convection in a saturated porous material of Darcy type. We allow a non-zero inertia term and also allow the permeability to be an anisotropic tensor. For a particular numerical result we consider the case of permeability tensor \(\mathbf{K}=\operatorname{diag} \{ {K_{\amalg},K_{\amalg},K_{\perp}(z) } \}\) . Linear instability results are calculated numerically and it is proved that the nonlinear energy stability bound is the same as the linear one.     

The Effective Conductivity of a Regular Composite Medium

The advantages of a Green's function method are exemplifiedby the analysis of steady potential flow through a homogeneousmedium containing a regular array of subdomains where a differentvalue of the conductivity prevails. It is shown, in particular,that the results previously obtained by Rayleigh for the squarearray of cylindrical obstacles with infinite conductivity arerecovered in a more efficient manner; and that the method issuitable for handling either widely dispersed or closely packedarrays.