MULTIPLE BOUNDARY LAYER PHENOMENA OF THE SOLUTION OF NONLINEAR HIGHER ORDER ELLIPTIC EQUATIONS WITH PERTURBATION BOTH IN BOUNDARY AND IN OPERATOR

1IntroductionInmanyproblemsofmechanicsandphysics,therefrequentlyappeartheellipticequationswiththederivativetermofthehigherordercontainingsmallparameters.Inengineering,thehigherorderellipticequationswithbothboundaryperturbationandoperatorperturbationoccurintheproblemsofcircularorellipticcoluxnnarshell(suchasthehullofasubmarineorthecolumnarshellroofetc.)withstiffeningribinthecirculnferentialdirection.Insuchcasesjthestudyofweldingstresscausedbyexternalpressureorself-weightduetoabnormalweldingposi…     

Boundary layers for compressible Navier-Stokes equations with outflow boundary condition

In this paper, we study the asymptotic relation between the solutions to the initial boundary value problem of the one-dimensional compressible full Navier-Stokes equations with outflow boundary condition and the associated Euler equations. We assume all the three characteristics to the corresponding Euler equations are all negative up to some small time, then we prove the existence and the stability of the boundary layers as long as the strength of the boundary layers is suitably small.     

一类四阶非线性奇摄动方程的边界层问题

本文研究了一类四阶非线性奇摄动方程的边界层问题,利用在左右边界层的两次匹配,得出了原问题解的一致有效的渐近表达式.这个结果是奇摄动理论在研究高阶微分方程中的一个应用.     

奇摄动高阶椭圆型方程解的多重边界层现象

研究了一类奇摄动2m阶椭圆型方程解的多重边层现象．利用比较定理得到解的一致有效的渐近展开式．     

A SINGULAR SINGULARLY-PERTURBED BOUNDARY VALUE PROBLEM

ＡＳＩＮＧＵＬＡＲＳＩＮＧＵＬＡＲＬＹ－ＰＥＲＴＵＲＢＥＤＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭ￥ＬｉｎＷｕｚｈｏｎｇ（林武忠）；ＷａｎｇＺｈｉｍｉｎｇ（汪志鸣）（ＥａｓｔＣｈｉｎａＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，上海华东师范大学，邮编：２０００...     

Vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system

In this paper, we study the vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system in a bounded domain. We first show the local existence of smooth solutions of the Euler/Allen-Cahn equations by modified Galerkin method. Then using the boundary layer function to deal with the mismatch of the boundary conditions between Navier-Stokes and Euler equations, and assuming that the energy dissipation for Navier-Stokes equation in the boundary layer goes to zero as the viscosity tends to zero, we prove that the solutions of the Navier-Stokes/Allen-Cahn system converge to that of the Euler/Allen-Cahn system in a proper small time interval. In addition, for strong solutions of the Navier-Stokes/Allen-Cahn system in 2D, the convergence rate is cν1/2.     

MULTIPLE LAYER BEHAVIOR OF BOUNDARY VALUE PROBLEM FOR SINGULARLY PERTURBED n-TH ORDER NONLINEAR DIFFERENTIAL EQUATION

In this paper,we study the singular perturbation of boundary value problem for a class of n-th order nonlinear differential equation,the solution is shown to exhibit multiple layer behavior.According to different layers and by introducing extended variable,we obtain the uniformly effective asymptotic expansion with different boundary layer correction term.     

二阶半线性奇摄动边值问题解的渐近行为

该文研究非法向双曲条件下的二阶半线性奇摄动边值问题解的渐近行为.利用边界层函数法,构造了区间端点处的代数型边界层,获得了问题的一致有效渐近解;利用微分不等式理论,证明了解的存在性以及渐近解与精确解之间的误差估计.通过一个典型的算例,验证了该文的理论结果.     

含双参数的三阶非线性两点边值问题的套层解(英文)

对含双参数且有套层解的一类三阶非线性微分方程的边值问题的渐近解做了估计 ,得到了任意次近似的一致有效的渐近展开式 .     

三阶非线性两点边值问题的奇摄动

本文借助不动点原理,对一类三阶非线性方程的边值问题的渐近解做了估计,得到了包括边界层在内的任意次近似的一致有效的渐近展开式。     

The Asymptotic Behavior and the Quasineutral Limit forthe Bipolar Euler-Poisson System withBoundary Effects and a Vacuum

In this paper, a one-dimensional bipolar Euler-Poisson system(a hydrodynamic model) from semiconductors or plasmas with boundary efects is considered. This system takes the form of Euler-Poisson with an electric field and frictional damping added to the momentum equations. The large-time behavior of uniformly bounded weak solutions to the initial-boundary value problem for the one-dimensional bipolar Euler-Poisson system is firstly presented. Next, two particle densities and the corresponding current momenta are verified to satisfy the porous medium equation and the classical Darcy’s law time asymptotically. Finally, as a by-product, the quasineutral limit of the weak solutions to the initial-boundary value problem is investigated in the sense that the bounded L∞entropy solution to the one-dimensional bipolar Euler-Poisson system converges to that of the corresponding one-dimensional compressible Euler equations with damping exponentially fast as t → +∞. As far as we know, this is the first result about the asymptotic behavior and the quasineutral limit for the one-dimensional bipolar Euler-Poisson system with boundary efects and a vacuum.     

在子区间上奇异摄动的一类边值问题解的渐近分析

研究了在子区间上奇异摄动的一类半线性二阶微分方程边值问题,用边界层函数法构造出问题的形式渐近解,借助微分不等式理论证明了渐近解的一致有效性.     

合成展开法应用于球壳对称弯曲的边界层问题

本文推广钱伟长在[5]中提出的合成展开法分析双参数边界层问题. 对于受均布荷载作用的球壳对称变形问题,其非线性平衡方程可以写成(2.3a),(2.3b):式中ε与δ是待定参数.当δ=1,ε是小参数时,这是第一边界层问题:当δ与ε都县小参数时.这是第二边界层问题. 对于上述问题,我们假定ε,δ和p满足ε3pδ=1-ε在这个条件下,应用推广的合成展开法,求出上述问题具有固定边界条件情况的渐近解.     

半线性奇摄动椭圆型方程边值问题的渐近解

本文运用了边界层函数构造了一类半线性奇摄动椭圆型方程边值问题解的渐近展开式，并证明了该展开式达到任一精度的一致有效性．     

Convergence to rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations

The main purpose of this paper is to study the asymptotic equivalence of the Boltzmann equation for the hard-sphere collision model to its corresponding Euler equations of compressible gas dynamics in the limit of small mean free path. When the fluid flow is a smooth rarefaction (or centered rarefaction) wave with finite strength, the corresponding Boltzmann solution exists globally in time, and the solution converges to the rarefaction wave uniformly for all time (or away from t=0) as ?→0. A decomposition of a Boltzmann solution into its macroscopic (fluid) part and microscopic (kinetic) part is adopted to rewrite the Boltzmann equation in a form of compressible Navier-Stokes equations with source terms. In this setting, the same asymptotic equivalence of the full compressible Navier-Stokes equations to its corresponding Euler equations in the limit of small viscosity and heat conductivity (depending on the viscosity) is also obtained.     

Global existence and asymptotic behavior of smooth solutions to a bipolar Euler–Poisson equation in a bound domain

In this paper, we present a bipolar hydrodynamic model from semiconductor devices and plasmas, which takes the form of bipolar isentropic Euler–Poisson with electric field and frictional damping added to the momentum equations. We firstly prove the existence of the stationary solutions. Next, we present the global existence and the asymptotic behavior of smooth solutions to the initial boundary value problem for a one-dimensional case in a bounded domain. The result is shown by an elementary energy method. Compared with the corresponding initial data case, we find that the asymptotic state is the stationary solution.     

关于非线性初边值问题的奇摄动(Ⅱ)

本文研究一类拟线性双曲—抛物型方程,具有变动边界的初边值问题的奇摄动:在某些条件成立,且ε充分小时,此问题的解具有以退化问题充分光滑解为首项的广义渐近展开式(Van der Corput意义),它在充分光滑解存在的区域Q={(x,t)|l₀(t)≤x≤l₁(t),0≤t≤T}上一致有效.其边界层存在于t=0附近.本文是工作[3]～[5]的进一步发展.     

A high order HODIE finite difference scheme for 1D parabolic singularly perturbed reaction-diffusion problems

This paper deals with the numerical approximation of the solution of 1D parabolic singularly perturbed problems of reaction-diffusion type. The numerical method combines the standard implicit Euler method on a uniform mesh to discretize in time and a HODIE compact fourth order finite difference scheme to discretize in space, which is defined on a priori special meshes condensing the grid points in the boundary layer regions. The method is uniformly convergent having first order in time and almost fourth order in space. The analysis of the uniform convergence is made in two steps, splitting the contribution to the error from the time and the space discretization. Although this idea has been previously used to prove the uniform convergence for parabolic singularly perturbed problems, here the proof is based on a new study of the asymptotic behavior of the exact solution of the semidiscrete problems obtained after the time discretization by using the Euler method. Some numerical results are given corroborating in practice the theoretical results.     

Note on the uniqueness of subsonic Euler flows in an axisymmetric nozzle

In this paper, we consider the uniqueness of globally subsonic compressible flows through an infinitely long axisymmetric nozzle. The flow is governed by the steady Euler equations and satisfies no-flow boundary conditions on the nozzle walls. We will show that for given mass flux and Bernoulli’s function in the upstream, the subsonic flow is unique in the class of all axisymmetric solutions, which possess the asymptotic behaviors at the far fields. This result extends the uniqueness of solutions in the previous paper Du and Duan (2011) [1].     

不动点定理在奇摄动非线性初值问题的套层解中的应用

Singularperturbationofnonlinearboundaryvalueproblemswhichhavedoubleboundarylayerpropertieswerediscussedbysomeauthorsusingthemethodofdifferentialinequali ties[1 ] -[3 ] .Nonlinearboundaryvalueproblemswhichhavenotdoubleboundarylayerpropertieswerediscussedbysomeauthorsusingthemethodofthefixed_pointtheorem[4] -[6 ] .Inthispa perwewillusethefixed_pointtheoremtodiscussthethirdorderdifferentialequationswhichhavedoubleinitiallayer.Weconsiderthefollowingproblem .ε4y =h(t,y,εy′,ε2 y″,ε) ,t∈ [0 ,T…