HOMOGENIZATION OF SOME LINEAR AND SEMILINEAR SCHRDINGER EQUATIONS WITHREAL POTENTIAL

1 IntroductionIn this paPer, we collsider the initial and boundaxy value probIem of linear and sentilillearScl1r5dinger equation as fOllows:l ie(x):5:i".,'<">"'>.'<",,'>,"=' 1:!\:?0t:)t>, (l1)wllere E > 0, fl C RN is a bounded open dolllain and T is an ar…     

MULTISCALE HOMOGENIZATION OF NONLINEAR HYPERBOLIC EQUATIONS WITH SEVERAL TIME SCALES

We study the multiscale homogenization of a nonlinear hyperbolic equation in a periodic setting.We obtain an accurate homogenization result.We also show that as the nonlinear term depends on the microscopic time variable,the global homogenized problem thus obtained is a system consisting of two hyperbolic equations.It is also shown that in spite of the presence of several time scales,the global homogenized problem is not a reiterated one.     

Homogenization of the Incompressible Navier-Stokes Fluid With Oscillation Coefficient

We study the homogenization of the incompressible Navier-Stokes equations with periodic oscillating coefficient in a bounded non-homogeneous media. To do that, we introduce a generalized compensate compactness result and a suitable class of test function to this problem. By passing the limit, we obtain the homogenized model of this problem.     

HOMOGENIZATION OF THE INCOMPRESSIBLE NAVIER-STOKES FLUID WITH OSCILLATION COEFFICIENT

We study the homogenization of the incompressible Navier-Stokes equations with periodic oscillating coefficient in a bounded non-homogeneous media. To do that, we introduce a generalized compensate compactness result and a suitable class of test function to this problem. By passing the limit, we obtain the homogenized model of this problem.     

Kantorovich distance in the martingale CLT and quantitative homogenization of parabolic equations with random coefficients

The article begins with a quantitative version of the martingale central limit theorem, in terms of the Kantorovich distance. This result is then used in the study of the homogenization of discrete parabolic equations with random i.i.d. coefficients. For smooth initial condition, the rescaled solution of such an equation, once averaged over the randomness, is shown to converge polynomially fast to the solution of the homogenized equation, with an explicit exponent depending only on the dimension. Polynomial rate of homogenization for the averaged heat kernel, with an explicit exponent, is then derived. Similar results for elliptic equations are also presented.     

Homogenization of linear and nonlinear transport equations

We study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields. Two types of homogenized equations are derived. For general n-dimensional linear and nonlinear problems, we derive homogenized equations by introducing additional independent variables to represent the small scales. For the two-dimensional linear transport equations, we derive effective equations for the averaged quantities. Such equations take the form of either a degenerate non-local diffusion equation with memory or a higher order hyperbolic equation. To study the nonlinear transport equations we introduce the concept of two-scale Young measure and extend DiPerna's method to prove that it reduces to a family of Dirac measures.     

Constructing zeros of accretive operators II

In this paper we study a homogenization problem for a time periodic boundary value problem concerning the quasi-stationary Maxwell equations with a non linear monotone magne tic characteristic. The main features of the problem are related to the vanishing of the conductivity inside each period so that the type of the equations is rapidly oscillating. The unknowns are a vector potential and a scalar potential. We show that the first one converges to zero up to terms of second order, while the second one converges to the solution of a suitable homogenized stationary equation (with time as a parameter). We show moreover that when the magnetic characteristic is linear and symmetric the second order terms in the asymptotic expansion of the vector potential can be identified and related to the time derivative of the limit scalar potential.     

A stability theorem of backward stochastic differential equations and its application

In this Note, we establish a stability theorem for backward stochastic differential equations, and we apply this theorem to study the homogenization of systems of semilinear parabolic partial differential equations.     

Homogenization of boundary value problems in perforated domains with the third boundary condition and the resulting change in the character of the nonlinearity in the problem

We study the homogenization problem for the Poisson equation in a periodically perforated domain with a nonlinear boundary condition for the flux on the cavity boundaries. We show that, under certain relations on the problem scale, the homogenized equations may have different character of the nonlinearity. In each case considered, we obtain estimates for the convergence of solutions of the original problem to the solution of the homogenized problem in the corresponding Sobolev spaces.     

Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

We describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved by the preconditioned conjugate gradient iteration with a convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). Using this solver, we numerically investigate the convergence of the representative volume element (RVE) method in stochastic homogenization that extracts the effective behavior of the random coefficient field. Our numerical experiments confirm the asymptotic convergence rate of systematic error and standard deviation in the size of RVE rigorously established in Gloria et al. The asymptotic behavior of covariances of the homogenized matrix in the form of a quartic tensor is also studied numerically. Our approach allows laptop computation of sufficiently large number of stochastic realizations even for large sizes of the RVE.     

带跳的分数倒向重随机微分方程及相应的随机积分偏微分方程

本文首次把Poisson随机测度引入分数倒向重随机微分方程,基于可料的Girsanov变换证明由Brown运动、Poisson随机测度和Hurst参数在(1/2,1)范围内的分数Brown运动共同驱动的半线性倒向重随机微分方程解的存在唯一性.在此基础上,本文定义一类半线性随机积分偏微分方程的随机黏性解,并证明该黏性解由带跳分数倒向重随机微分方程的解唯一地给出,对经典的黏性解理论作出有益的补充.     

A Hilbert space approach to homogenization of linear ordinary differential equations including delay and memory terms

We present an abstract approach to homogenization in a Hilbert space setting. Related compactness results are obtained. Moreover, the homogenized equations may be computed explicitly, if periodicity is imposed. Examples for the applicability of our homogenization result for linear ordinary (integro‐)differential equations are given. Copyright © 2012 John Wiley & Sons, Ltd.     

HOMOGENIZATION,SYMMETRY, AND PERIODIZATION IN DIFFUSIVE RANDOM MEDIA

We present a systematic study of homogenization of diffusion in random media with emphasis on tile-based random microstructures.We give detailed examples of several such media starting from their physi...     

Hopf bifurcation in a size-structured population dynamic model with random growth

This paper is devoted to the study of a size-structured model with Ricker type birth function as well as random fluctuation in the growth process. The complete model takes the form of a reaction-diffusion equation with a nonlinear and nonlocal boundary condition. We study some dynamical properties of the model by using the theory of integrated semigroups. It is shown that Hopf bifurcation occurs at a positive steady state of the model. This problem is new and is related to the center manifold theory developed recently in [P. Magal, S. Ruan, Center manifold theorem for semilinear equations with non-dense domain and applications to Hopf bifurcation in age-structured models, Mem. Amer. Math. Soc., in press] for semilinear equation with non-densely defined operators.     

广义双尺度收敛和具分层周期结构系数的偏微分方程的均匀化

我们给出了广义双尺度收敛的概念。用广义双尺度收敛的理论我们研究了某些具分层周期结构系数的线性和半线性偏微分方程。     

Optimality Conditions for Semilinear Hyperbolic Equations with Controls in Coefficients

An optimal control problem for semilinear hyperbolic partial differential equations is considered. The control variable appears in coefficients. Necessary conditions for optimal controls are established by method of two-scale convergence and homogenized spike variation. Results for problems with state constraints are also stated.     

Homogenization of boundary layer of pseudo-plastic fluid in the presence of rapidly oscillating external forces

In this paper, we study the homogenization problem for equations of magnetohydrodynamic boundary layer of pseudo-plastic fluid. It is assumed that the external flow velocity and the external magnetic field are described by oscillating functions and the frequency depends on a small parameter. In von Mises variables and in Cartesian variables, we construct the homogenized problem, establish strong convergence of solutions in a special norm, and estimate the rate of this convergence. We show that in von Mises variables the convergence rates in different norms are of different orders of smallness.     

Low-volume-fraction limit for polymer fluids

We study a stationary, purely viscous polymer flow through a porous medium modelled as a periodic array of cells consisted of a fluid part and a solid one. Solid parts of the domain present impermeable obstacles, whose impact on fluid flow may be seen as a slowing factor through averaged quantities such as the permeability function, obtained by the homogenization process. In that way, the influence of the microstructure is implemented in the homogenized equations through a kind of nonlinear Darcy's law. Our goal is to find more explicitly the dependence of the permeability function on the size η of the obstacle in the unit cell and the so-called low-volume-fraction limit. Main difficulties arise from the nonlinear character of the power-law viscosity and the apparent weak convergence of the solutions involved.     

Weak turbulence plasma induced by two-scale homogenization

This paper is devoted to the homogenization of the quasilinear theory of the plasma turbulence described by the Vlasov–Poisson system. It is shown that the homogenization limit, in the sense of two-scale limit, of the distribution function satisfies the linear Vlasov–Poisson equations. Moreover, the limit distribution function can be decomposed into the mean and the fluctuation parts and the mean part (the equilibrium distribution function) is shown to be the solution of the nonlocal quasilinear velocity-space diffusion equation. We also investigate the Landau damping from the point of view of homogenization through the two-scale limit.     

Stochastic-periodic homogenization of Maxwell’s equations with linear and periodic conductivity

Stochastic-periodic homogenization is studied for the Maxwell equations with the linear and periodic electric conductivity. It is shown by the stochastic-two-scale convergence method that the sequence of solutions to a class of highly oscillatory problems converges to the solution of a homogenized Maxwell equation.