Solvability of Control Problems for Stationary Equations of Magnetohydrodynamics of a Viscous Fluid

We consider the boundary-value problem for the stationary equations of magnetohydrodynamics of a viscous incompressible fluid with nonhomogeneous boundary conditions for the velocity and electromagnetic field. We study global solvability of this problem and establish some sufficient conditions for uniqueness of its solution. We state control problems for the model of magnetohydrodynamics under consideration, study their solvability, give and examine optimality systems for both arbitrary and particular quality functionals.     

Waves in an Inhomogeneous Tube with a Flowing Two-Phase Fluid

Results of theoretical and mathematical justification of the problem on a pulsating flow of a two-phase barotropic bubbly fluid enclosed in an elastic semi-infinite cylindrical tube inhomogeneous along its length are presented. Linear one-dimensional equations are used. It is assumed that the tube is rigidly attached to the surrounding medium and therefore its displacement in the axial direction is absent. At infinity, the tube material is assumed to be homogeneous. To describe the pressure, flow rate, and displacement of the fluid, a pulsating pressure is given at the tube end. The problem stated is reduced to a singular Sturm-Liouville boundary-value problem, which in turn is reduced to a Volterra-type integral equation. This equation is solved by the method of successive approximations. By assuming that the corresponding potential is integrable, it is proved that these approximations converge to the exact solution of the problem. It is shown that this assumption also covers the very important practical case of piecewise inhomogeneity. For numerical realization, we consider a homogeneous tube with flowing water containing a small amount of bubbles. The effect of the volume content of bubbles on wave characteristics is revealed. In particular, it is stated that, for the oscillation regime selected, an increased bubble volume content decreases the wave velocity and considerably increases the flow speed (rate).     

基于Roe格式的可压与不可压流的统一计算方法

摘要：以Navier-Stokes方程为基础，基于有限体积的时间推进的预处理技术．提出了一个可以用来求解可压与不可压流场的统一的计算方法，原始变量选用压力、速度与温度，通过矩阵变换与重构，使得对流项系数矩阵在可压与小可压条件下都不会奇异．将可压与不可压流场的计算方法统一起来。采用Roe格式计算对流通量，采用中心差分格式计算扩散通量．算例表明，该方法可以进行高Mach数、中等Mach数、低Mach数及不可压流场的计算。由于采用了Roe格式，该方法还可以捕获不连续流场的间断面。     

Boundary Asymptotic Analysis for an Incompressible Viscous Flow: Navier Wall Laws

We consider a new way of establishing Navier wall laws. Considering a bounded domain Ω of R ^N , N=2,3, surrounded by a thin layer Σ _ε , along a part Γ₂ of its boundary ∂Ω, we consider a Navier-Stokes flow in Ω∪∂Ω∪Σ _ε with Reynolds’ number of order 1/ε in Σ _ε . Using Γ-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier law involving a matrix of Borel measures having the same support contained in the interface Γ₂. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider an optimal control problem within this context.     

两个旋转球之间粘性不可压缩流动的Lorenz系统

两个不同角速度旋转球之间粘性流动问题是地球外部大气流动的简化模型.通过引入球Bessel函数的有理表达式,得到Stokes算子特征值与特征函数的有理表达形式.利用Stokes算子特征函数作为基函数系,对两个旋转球间流动问题进行谱Galerkin逼近.由三模态的Glerkin逼近方程得到—个类Lorenz系统,我们对此系统进行分歧问题和吸引子的讨论,从而得到原问题的稳定性判定.     

边界控制下一阶半线性脉冲调宽采样系统的稳态控制

首先证明了一类一阶非线性系统在周期边界控制下存在全局稳定的周期解.同时,用该结论证明了一类一阶非线性脉冲调宽采样系统存在稳态控制.     

对波动方程初边值问题边界条件齐次化函数的一个注解

基于传统的齐次化边界条件方法,采用傅里叶级数法讨论了波动方程初边值问题第一类非齐次边界条件齐次化函数问题,分析表明:对同一定解问题,在不同齐次化函数下的解在适定意义下是等价的.     

一类由抛物-双曲耦合组支配的边界最优控制问题

本文讨论了一类由抛物-双曲耦合组支配的第三边值最优控制问题.证明了该问题最优控制的存在性及其bang-bang性质.     

弱粘性流体中垂直激励表面波的阻尼效应

在竖直振动的圆柱形容器中,将Navier-Stokes方程线性化,利用两时间尺度奇异摄动展开法研究了弱粘性流体的单一自由面驻波运动．整个流场被分为外部势流区和内部边界层区两部分,对两部分区域分别求解,得到包含阻尼项和外驱动影响的线性振幅方程．利用稳定性分析,得到形成稳定表面波的条件,给出了临界曲线．此外,还获得了阻尼系数的解析表达式．最后,将线性阻尼加到理想流体条件下所得到的色散关系中对其进行修正,理论结果证明修正后的驱动频率更加接近实验的结果．通过计算发现,当驱动的频率较低时,流体的粘性对表面波模式选择有重要影响,而表面张力的影响不明显;但当驱动频率较高时,流体的表面张力起主要作用,而流体的粘性影响甚小．     

一个与硅氧化有关的粘性可压缩流体的自由边界问题

本文研究一个描述硅的氧化过程的自由边界问题．它的数学模型是一个可压缩的Ｎａｖｉｏｒ－Ｓｔｏｋｅｓ方程与一个抛物方程以及一个双曲方程的耦合，其中在自由边界上存在表面张力并且密度方程是非齐次的．本文将证明只要已知数据满足相容性条件，则上述问题有唯一局部强解．     

Asymptotic Stability of a Stationary Flowing Regime of an Ideal Incompressible Fluid

We give sufficient conditions for asymptotic stability of a stationary solution to a flowing problem of a homogeneous incompressible fluid through a given planar domain. We consider a planar problem for the Euler equation and boundary conditions for the curl and the normal component of the velocity; moreover, the latter is given on the whole boundary of the flow domain and the curl is given only on the inlet part of the boundary. We establish asymptotic stability of a stationary flow (in linear approximation), assuming it to have no rest points and to satisfy some smallness condition which means that the perturbations leave the flow domain before they become to affect the main flow. In particular, we prove asymptotic stability for an arbitrary stationary flow in a rectangular canal close to the Couette flow without rest points. Moreover, we show that stability of the main flow in the L ₂-norm under curl perturbations implies its stability in higher-order norms depending, for example, on the derivatives of the curl.     

由发展算子确定的具有控制不等式约束的LQCP的最优反馈控制

研究由发展算子定义的具有控制函数积分不等式约束的线性二次控制问题（LQCP）,借助无约束线性二次控制系统推导出了最优控制的反馈形式。     

Solvability of the Boundary-Value Problem for a Mathematical Model of Steady-State Flows of Nonlinear-Viscous Fluids

We establish an existence theorem for weak solutions of the boundary-value problem for steady-state equations describing both laminar and turbulent flows of nonlinear-viscous fluids.     

一类可修复计算机系统的最优控制问题

利用Banach空间的相关理论,讨论了一类可修复计算机系统稳态解的最优控制问题,并证明了最优控制元的存在性与唯一性.结果表明,计算机系统的稳态可用度在有限时间内总能到达其期望值.     

Optimal Control of PDEs with Regularized Pointwise State Constraints

This paper addresses the regularization of pointwise state constraints in optimal control problems. By analyzing the associated dual problem, it is shown that the regularized problems admit Lagrange multipliers in L²-spaces. Under a certain boundedness assumption, the solution of the regularized problem converges to the one of the original state constrained problem. The results of our analysis are confirmed by numerical tests. Supported by the DFG Research Center “Mathematics for key technologies” (FZT 86) in Berlin.     

Use of Augmented Lagrangian Methods for the Optimal Control of Obstacle Problems

We investigate optimal control problems governed by variational inequalities involving constraints on the control, and more precisely the example of the obstacle problem. In this paper, we discuss some augmented Lagrangian algorithms to compute the solution.     

固壁近旁Stokes流中粘性液滴的运动和变形

发展了一种模拟固壁近旁轴对称Stokes流中粘性液滴的运动和变形及直接计算固壁上应力的边界积分方法．用此方法对不同的液滴-固壁初始相对间距、粘度比、表面张力和浮力联合参数以及环境流动参数情况进行了数值实验．数值结果显示,由于环境流动和浮力的作用,随着时间的推进,液滴在轴向压缩,在径向拉伸．当环境流动的作用弱于浮力作用时,随着时间的推移,液滴上升并向上弯,固壁上由液滴运动所引起的应力不断减小．当环境流动的作用强于浮力作用时,随着时间的推移,液滴变得越来越扁．在这种情形,当大初始间距时,壁面上的应力随液滴的演变而增大;当小初始间距时,由环境流动、浮力及壁面对流动的较强作用的联合影响,此应力随液滴的演变而减小．由于液滴运动所引起的壁面应力的有效作用仅限于对称轴附近的一个小范围内,且此范围随液滴与固壁的初始间距增大而增大．应力的大小随初始间距增大而大为减小．表面张力对液滴变形有阻止作用．液滴粘性会减小液滴的变形和位置迁移．     

存在滑移时两圆球间的幂律流体挤压流动

基于Reynolds润滑理论分析了壁面滑移对任意圆球颗粒间幂律流体的挤压流动的影响。研究表明有壁面滑移时挤压流动的粘性力可通过引进本文定义的滑移修正系数分离出无滑移解。推导出的挤压力滑移修正系数是一积分表达式，依赖于滑移参数、幂律指数、球间隙和积分上限。一般地壁面滑移导致粘性力减小，粘性力的减小量随幂律指数的增大而增大，表明壁面滑移对剪切增稠流变材料有更大的影响；粘性力的减小量还随着滑移参数的增大而增加，而这恰与假设相符合；粘性力的减小量又随球间隙减小或积分上限的增大（从液桥情况到完全浸渍）而减小直到趋于常数，这一特性在离散元模拟时可以有效地减少计算量。     

Necessary conditions for an extremum in the calculus of variations with nonsmooth phase variables

Necessary conditions for an extremum in a general isoperimetric problem not involving partial derivatives with respect to phase variables are obtained. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 922–930, June, 1997. Translated by N. K. Kulman