WELL-POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INVISCID COMPRESSIBLE ADIABATIC FLUID ＊

Introduction Itisreasonabletoconsiderarealfluidasanidealoneinfluidmechanicsundermany conditions.Forinstance,ingeneral,forthedistributionofthefluidfieldaroundtheaerocraft,mostpartofthefluidfieldmayberegardedasidealfluidexceptforasmallpartwhereeffects ofviscousandheatconductioninthethinlayernearthesurfacemustbeconsidered.Evenif thefluidiscompletelysupposedasidealonethroughoutthefluidfield,thequitereasonable resultsarealsogained,thereforestudyingidealfluidhasnotonlytheoreticalsignificancebut also…     

A moving grid method applied to one-dimensional non-stationary flame propagation

The paper presents applications of a moving grid method to the combined problem of ignition and premixed flame propagation in a closed vessel. This method belongs to the general class of adaptive grid techniques for the numerical integration of evolutionary partial differential equations and is based on the method of lines with variable node position. In the present case the motion of the grid and the solution of the partial differential equations are strongly coupled by an implicit formulation. The problem is reduced to an initial value problem for a stiff differential-algebraic system. The continuously moving grid is determined by the equidistribution of a positive function which depends on the solution of the partial differential equations. A differential-algebraic system solver is used for the time integration of the initial value problem. The numerical results of the test problems demonstrate the computational efficiency and the capability of the method to resolve the main features of the solution.     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID (Ⅰ)

1　IntroductionandSignsForthestudyofwellposednessontheinitialorboundaryproblemsabouttheEulerequations,therearemanyimportantresultsgivenbydifferentmethodsindifferentfunctionclasses[1- 3].Andthereislessdiscussiononitsillposedness[4 ,5 ].Inthedifferentialfunctionclasses,BaouondiandGoulaouicprovedthattheCauchyproblemonthecompactmanifoldofEulerequationshasuniquesolutionwellexpectaconstant[4 ].InthebooktitledSolutionsAnalytiquesdeQuelquesEquationsauxDerivesPartiellesenMecaniquedesFluides,thest…     

Blow-up of solution for a generalized Boussinesq equation

This paper studies the initial boundary value problem for a generalized Boussinesq equation and proves the existence and uniqueness of the local generalized solution of the problem by using the Galerkin method.Moreover,it gives the sufficient conditions of blow-up of the solution in finite time by using the concavity method.     

Initial value problem for a class of fourth-order nonlinear wave equations

In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.     

Numerical modeling of multiphase first-contact miscible flows. Part 1. Analytical Riemann solver

In this series of two papers, we present a front-tracking method for the numerical simulation of first-contact miscible gas injection processes. The method is developed for constructing very accurate (or even exact) solutions to one-dimensional initial-boundary-value problems in the form of a set of evolving discontinuities. The evolution of the discontinuities is given by analytical solutions to Riemann problems. In this paper, we present the mathematical model of the problem and the complete Riemann solver, that is, the analytical solution to the one-dimensional problem with piecewise constant initial data separated by a single discontinuity, for any left and right states. The Riemann solver presented here is the building block for the front-tracking/streamline method described and applied in the second paper.     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅰ)

The ill posed initial value problem of the Euler equations and the formal solvability of ill posed problem based on stratification theory are discussed. For some ill posed initial value problems, the existence conditions of formal solutions and the methods of how to construct a formal solution are given. Finally, an example is given to discuss the ill posedness of the initial value problem on hyper plane {t=0} in R⁴, and explain that the problem has more than one solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID (Ⅱ)

IntroductionForthestudyofwellposednessontheinitialorboundaryproblemsabouttheEulerequations,therearemanyimportantresultsgivenbydifferentmethodsindifferentfunctionclasses[1]- [4 ].Thepaper (Ⅰ )oftheauthordiscussedtheill_posednessoftheCauchyproblemswhichsatisfythesameconditionsoftheinitialvalueproblematthehypersurfacet=t( 0 ) +g(x) ,g(x) ≠constant,g(0 ) =0 R4,andgavetwodifferentformalsolutions[5 ].Italsogavetheformulasforcomputation .Inthepresentpaper,twokindsofillposednessoftheEulerequation…     

A Meshless Solution Technique for the Solution of 3D Unsaturated Zone Problems,Based on Local Hermitian Interpolation with Radial Basis Functions

Recent developments in meshless numerical methods have led to algorithms that can be used to solve arbitrarily large problems without the support of a connected mesh, and without the computational cost and numerical ill-conditioning issues usually associated with such solution techniques. This work applies the Local Hermitian Interpolation (LHI) method, based on local interpolation with Radial Basis Functions (RBFs), to the solution of 3D unsaturated porous media problems. The proposed implementation is capable of handling real soil properties, provided either as an analytical function or as a series of pointwise measurements. The technique is implemented with implicit and explicit timestepping, and is validated against two transient Richards’ equation models, of which one has a known analytical solution. In addition, a real-world infiltration problem based on a saturated–unsaturated formulation is modelled, using a realistic variation of soil properties with water-pressure.     

Analytical Solution for Sliding Rounded-Edge Contact

Reliable analysis of the local stress fields in the vicinity of sliding frictional contacts of engineering components is a pre-requisite for the reliable assessment of structural integrity. In this paper we present the use of Muskhelishvili potentials to derive an analytical solution for a semi-infinite punch with a rounded edge pressed against a half-pane substrate, and a general numerical solution for a kind of Cauchy integral involved in the analytical contact solution. Using these solutions, the effect of the friction coefficient on the normal traction distribution is investigated. Numerical results show that the peak normal traction value is altered in comparison with the frictionless case solution, but this variation is mild (less than 5%), provided the friction coefficient does not exceed about 0.6. Finally, the adaptation of the analytical result to the solution of practical contact problems is addressed.     

On constructing a Green’s function for a semi-infinite beam with boundary damping

The main aim of this paper is to contribute to the construction of Green’s functions for initial boundary value problems for fourth order partial differential equations. In this paper, we consider a transversely vibrating homogeneous semi-infinite beam with classical boundary conditions such as pinned, sliding, clamped or with a non-classical boundary conditions such as dampers. This problem is of important interest in the context of the foundation of exact solutions for semi-infinite beams with boundary damping. The Green’s functions are explicitly given by using the method of Laplace transforms. The analytical results are validated by references and numerical methods. It is shown how the general solution for a semi-infinite beam equation with boundary damping can be constructed by the Green’s function method, and how damping properties can be obtained.     

文克尔地基上阶梯式矩形板弹性曲面微分方程的解析解

本文应用阶跃函数建立了文克尔地基上阶梯式矩形薄板弹性曲面微分方程,并用初参法和作者引入的W运算符获得了该微分方程解的解析表达式。文中举例说明了本文方法的应用。     

各向异性介质中SH波引起的裂纹扩展

本文利用Green函数法,求解各向异性介质中半无限长裂纹在SH波作用下,以任意速度扩展的问题。首先,利用Laplace变换和Cagniard-de Hoop反演法求解各向异性介质中反平面问题的Green函数,并利用它建立了求解裂纹扩展问题的积分方程。因为方程为Abel型的,所以可得到在SH波作用下,半无限长裂纹扩展问题的解析解。还可求得裂纹端点附近的应力和裂纹表面上位移的表达式。并对裂纹端点附近的奇异性进行讨论。最后讨论了裂纹尖端附近任一点的能量关系。并应用Griffith的能量准则,对裂纹扩展规律进行了讨论。     

Global existence of solutions to the periodic initial value problems for two-dimensional Newton-Boussinesq equations

A class of periodic initial value problems for two-dimensional Newton- Boussinesq equations are investigated in this paper. The Newton-Boussinesq equations are turned into the equivalent integral equations. With iteration methods, the local existence of the solutions is obtained. Using the method of a priori estimates, the global existence of the solution is proved.     

双相介质波动方程孔隙率反演的同伦方法

从材料响应的理论合成应与实际测量数据相拟合这一出发点，将双相介质波劝方程参数的反演问题转化为非线性算子方程的零点求解问题，从而应用一种大范围收敛的同伦方尘土注来解非线性算子方程，并把这种方法用于Simon(1984)给出的具有解析的一维双相介质模型的数值模拟，最后的数值结果表明，给出的算法是十分有效的。     

Stability for basic system of equations of atmospheric motion

The topological characteristics for the basic system of equations of atmospheric motion were analyzed with the help of method provided by stratification theory. It was proved that in the local rectangular coordinate system the basic system of equations of atmospheric motion is stable in infinitely differentiable function class. In the sense of local solution, the necessary and sufficient conditions by which the typical problem for determining solution is well posed were also given. Such problems as something about ＂speculating future from past＂ in atmospheric dynamics and how to amend the conditions for determining solution as well as the choice of underlying surface when involving the practical application were further discussed. It is also pointed out that under the usual conditions, three motion equations and continuity equation in the basic system of equations determine entirely the property of this system of equations.     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅱ)

The solvability of the Euler equations about incompressible inviscid fluid based on the stratification theory is discussed. And the conditions for the existence of formal solutions and the methods are presented for calculating all kinds of ill-posed initial value problems. Two examples are given as the evidences that the initial problems at the hyper surface does not exist any unique solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

Extremum problems of boundary control for steady equations of thermal convection

An inverse extremum problem of boundary control for steady equations of thermal convection is considered. The cost functional in this problem is chosen to be the root-mean-square deviation of flow velocity or vorticity from the velocity or vorticity field given in a certain part of the flow domain; the control parameter is the heat flux through a part of the boundary. A theorem on sufficient conditions on initial data providing the existence, uniqueness, and stability of the solution is given. A numerical algorithm of solving this problem, based on Newton’s method and on the finite element method of discretization of linear boundary-value problems, is proposed. Results of computational experiments on solving extremum problems, which confirm the efficiency of the method developed, are discussed.     

区域脉冲载荷下二维Lamb问题的精确求解

采用积分变换方法,并利用两类积分公式克服反变换求解的困难,求得了区域脉冲载荷下一个二维Lamb问题的代数形式的精确解.基于该分段函数形式的代数结果,纵波、横波、Rayleigh波等应力波成分在弹性表面的激发和传播过程得到详尽分析,其中很多结论是已有的解析积分结果或者数值计算结果不曾得到的.     

用初参数法解C型波纹管在子午面内整体弯曲

波纹管是一种呈波纹状的管形薄壁结构,作为弹性敏感元件、密封元件和位移补偿器,在现代工业、国防和民用设备中的应用日益广泛.波纹管的计算,特别是在非轴对称受力和变形方面亟待完善.本文讨论了C型波纹管在子午面内整体弯曲的应力分布和端面角位移.从柔性圆环壳在子午面内整体弯曲的复变量方程出发,化边值问题为初值问题,用S.Gill法,得到了该波纹管的数值解,适用于0＜α＜1的范围.通过3个例子,将本文的数值解和已有的细环壳一般解(解析解)作了比较.结果表明,两者基本一致,并且,当环壳较细(α较小)时,两种计算结果更为接近.