Global solutions of the Navier-Stokes equations in a uniformly rotating space

We consider the Cauchy problem for the Navier-Stokes system of equations in a three-dimensional space rotating uniformly about the vertical axis with the periodicity condition with respect to the spatial variables. Studying this problem is based on expanding given and sought vector functions in Fourier series in terms of the eigenfunctions of the curl and Stokes operators. Using the Galerkin method, we reduce the problem to the Cauchy problem for the system of ordinary differential equations, which has a simple explicit form in the basis under consideration. Its linear part is diagonal, which allows writing explicit solutions of the linear Stokes-Sobolev system, to which fluid flows with a nonzero vorticity correspond. Based on the study of the nonlinear interaction of vortical flows, we find an approach that we can use to obtain families of explicit global solutions of the nonlinear problem.     

Finite-amplitude long-wave instability of Bingham liquid films

The long-wave perturbation method is employed to investigate the weakly nonlinear hydrodynamic stability of a thin Bingham liquid film flowing down a vertical wall. The normal mode approach is first used to compute the linear stability solution for the film flow. The method of multiple scales is then used to obtain the weak nonlinear dynamics of the film flow for stability analysis. It is shown that the necessary condition for the existence of such a solution is governed by the Ginzburg–Landau equation. The modeling results indicate that both the subcritical instability and supercritical stability conditions can possibly occur in a Bingham liquid film flow system. For the film flow in stable states, the larger the value of the yield stress, the higher the stability of the liquid film. However, the flow becomes somewhat unstable in unstable states as the value of the yield stress increases.     

Nonlinear stability characterization of thin Newtonian film flows traveling down on a vertical moving plate

The paper presents both the linear and nonlinear stability theories for the characterization of thin Newtonian film flows traveling down along a vertical moving plate. The linear model is first developed to characterize the flow behavior. After showing the inadequacy of the linear model in representing certain flow characteristics, the nonlinear kinematics model is then developed to represent the system. The long-wave perturbation method is employed to derive the generalized kinematic equations with free film surface condition. The linear model is solved by using the normal mode method for three different, namely, the quiescent, up-moving and down-moving, moving conditions. Subsequently, the elaborated nonlinear film flow model is solved by the method of multiple scales. The modeling results clearly indicate that both subcritical instability and supercritical stability conditions are possible to occur in the film flow system. The effect of the down-moving motion of the vertical plate tends to enhance the stability of the film flow.     

The stability of periodic solutions of discontinuous systems that intersect several surfaces of discontinuity

Systems of differential equations with discontinuous right-hand sides are considered, specifically investigating periodic solutions which simultaneously intersect two or more surfaces of discontinuity. It is shown that the Poincaré mapping along phase trajectories of the system in the neighbourhood of a fixed point, corresponding to periodic motion, is in general piecewise-differentiable: this neighbourhood divides into several sectors in which the Jacobians are different. For such mappings, theorems of stability in the first approximation [1] are not applicable, and one has to devise new stability criteria. Several necessary conditions for stability are obtained, as well as sufficient conditions. The results are used to investigate symmetric modes of motion of a vibro-impact system with two impact pairs.     

流体层流运动稳定性中的Ляпунов方法

本文是[1]的继续.在本文中,利用[1]的结果我们证明了,对于流体的层流运动稳定性而言,在线性化问题中,按特征值定义与按扰动能量定义二者是完全等价的,从外,借助于Ляпунов方法,我们又证明了,如果线性化问题是渐近稳定的,当考虑非线性影响时,只要扰动能量足够小,则仍然是渐近稳定的.     

A Higher-Order Internal Wave Model Accounting for Large Bathymetric Variations

A higher-order strongly nonlinear model is derived to describe the evolution of large amplitude internal waves over arbitrary bathymetric variations in a two-layer system where the upper layer is shallow while the lower layer is comparable to the characteristic wavelength. The new system of nonlinear evolution equations with variable coefficients is a generalization of the deep configuration model proposed by Choi and Camassa [ 1 ] and accounts for both a higher-order approximation to pressure coupling between the two layers and the effects of rapidly varying bottom variation. Motivated by the work of Rosales and Papanicolaou [ 2 ], an averaging technique is applied to the system for weakly nonlinear long internal waves propagating over periodic bottom topography. It is shown that the system reduces to an effective Intermediate Long Wave (ILW) equation, in contrast to the Korteweg-de Vries (KdV) equation derived for the surface wave case.     

Bifurcation to Mean Flows in Convection

Convection with a strip plate in the middle is studied in this paper. Simultaneous instability of two convection modes of different vertical structures with a same horizontal wave number is possible in this system. It is found that the interaction of these two modes generates mean flows similar to those observed by Krishnamurti and Howard [9] in a turbulent convection experiment. Coupled nonlinear equations are derived for the amplitudes of the two modes. Traveling wave solutions and more complicated time-dependent solutions are also found near the onset of convection.     

Une méthode de décomposition en temps avec des schémas dʼintégration réversible pour la résolution de système dʼéquations différentielles ordinaires

We propose a time domain decomposition method that breaks the sequentiality of the integration scheme for systems of ODE. Under the condition of differentiability of the flow, we transform the initial value problem into a well-posed boundary values problem using the symmetrization of the interval of time integration and time-reversible integration scheme. For systems of linear ODE, we explicitly construct the block tridiagonal system satisfied by the solutions at the time sub-intervals extremities. We then propose an iterative algorithm of Schwarz type for updating the interfaces conditions which can extend the method to systems of nonlinear ODE.     

Study of vortex breakdown in a stratified fluid

An analysis shows that nonsmooth solutions have to be considered. Weak solutions to the Euler equations describing an incompressible stratified fluid under gravity are defined and studied. The study makes use of a wave energy functional proposed for the nonlinear equations. It is shown that the Euler equations are insufficient for stating a well-posed generalized problem. Additional conditions based on physical considerations are proposed. One condition is energy conservation, and the other is a constraint imposed on the density, which is required for stability. A numerical method is developed that is used to analyze how wave breakdown in a stratified fluid depends on stratification. The numerical results are in satisfactory agreement with experiments.     

On strongly stable normal operators in Krein spaces

For bounded normal operators in Krein spaces we give a necessary and sufficient condition for strong stability. The same result for unitary operators was obtained by M.G.Krein [1] (see also [2]). For selfadjoint operators we refer to the papers of P.Jonas, H.Langer [3] and H.Langer [4].     

Importance of Tollmien's Counter Example

Efforts to construct a general theoretical basis containing the essential features of Tollmien's counter example to the sufficiency of Rayleigh's theorem on point of inflexion have resulted in the determination of a pair of upper bounds of the rate of growth of arbitrary unstable disturbances; whereas, the necessary condition of the existence of these upper bounds have provided access to a sufficient condition of stability in its simplest form in the equilibrium of homogeneous incompressible inviscid parallel shear flows that are not known as yet and go beyond the works of Rayleigh [1], Tollmien [2], Friedrichs [3], Fjortoft [4], Hoiland [5], Howard [6, 7], Hickernell [8], and Banerjee et al. [9]. An alternative proof of the result that a wide class of such flows could be made stable by bringing the boundaries sufficiently close, although the flow has a point of inflexion inside the domain of flow with the Fjortoft's criterion satisfied, which is derived by Drazin and Howard [10] from variational formulation of the problem follows as an outcome of the expressions of these upper bounds. The counter example has played the role of a forerunner for much of the development that followed in its wake after 1935, and the present succession of papers is especially undertaken to investigate the trail left behind by the counter example and, it is hoped, to arrive at a necessary and sufficient condition of stability in its simplest form, which is still missing in the literature on the subject.     

Analysis and Optimal Boundary Control of a Nonstandard System of Phase Field Equations

We investigate a nonstandard phase field model of Cahn-Hilliard type. The model, which was introduced in [16], describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been studied recently in [5], [6] for the case of homogeneous Neumann boundary conditions. In this paper, we investigate the case that the boundary condition for one of the unknowns of the system is of third kind and nonhomogeneous. For the resulting system, we show well-posedness, and we study optimal boundary control problems. Existence of optimal controls is shown, and the first-order necessary optimality conditions are derived. Owing to the strong nonlinear couplings in the PDE system, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional will be of standard type.     

A discrete solenoidal finite difference scheme for the numerical approximation of incompressible flows

Summary For a well known class of finite difference schemes for approximating incompressible flows it is shown that the condition of discrete incompressibility can be incorporated into the discrete space. This simplifies the structure of the linear or nonlinear discrete systems and reduces the number of unknowns.     

The effect of a high-frequency progressive vibration on the convective instability of a two-layer fluid

The influence of a high-frequency progressive vibration on the onset of thermal convection in a two-layer system of viscous immiscible fluids is investigated. The interface is deformable, the outer walls are rigid, and heat-transfer conditions of a general form are assigned on them. The starting equations are taken in the generalized Oberbeck–Boussinesq approximation. An averaging method is employed. It is shown that the averaged problem contains a vibrogenic external force and vibrogenic stresses that are proportional to the square of the amplitude of the vibration rate. A quasi-equilibrium solution that satisfies the closure condition is found, and its stability is investigated. It is established that, unlike the case of a single-layer fluid, the horizontal component of the vibration influences the onset of convection and have a destabilizing effect. The vertical component stabilizes the two-layer system by increasing the surface tension. The long-wavelength asymptotic is investigated. Calculations are performed for the silicone oil–Fluorinert and acetonitrile–n-hexane systems.     

On the spectral properties of three-layer difference schemes for parabolic equations with nonlocal conditions

We consider three-layer difference schemes for a one-dimensional linear parabolic equation with nonlocal integral conditions. A three-layer scheme is written out in an equivalent form of a two-layer scheme. We analyze the dependence of the spectrum of the difference operator on the parameters occurring in the integral conditions. We derive stability conditions for the original three-layer scheme in a specially defined energy norm.     

The legendre condition in optimum problems of supersonic gasdynamics : PMM vol. 39, n 6, 1975, pp. 1032–1042

The necessary Legendre condition for problems of optimum (in the sense of minimum wave drag) supersonic flow past bodies is obtained. Plane and axisymmetric flows are considered on the assumption of imposition of isoperimetric constraints of a general form. Shock-free flows and flows with attached shock waves are investigated. The method here proposed is used for deriving the second order condition in the particular case when it is possible to pass to the reference contour, and which has been earlier obtained by Shmyglevskii [1] and then by Guderley and others [2].     

弹性弦Dirichlet边界反馈控制的镇定与Riesz基生成

本文通过一端固定 ,一端 Dirichlet边界控制的一维波动方程说明系统是 Salamon- W eiss意义下适定和正则的 .由此说明 ,由 J.L.Lions引入的用于研究双曲方程精确可控性的 H ilbert唯一性方法是控制论中著名的对偶原理 .我们讨论了系统的指数镇定及闭环系统的广义本征函数生成 Riesz基和谱确定增长条件 .我们希望通过本文使读者对目前线性偏微分控制理论的一个新动向有一基本的了解 .     

A study of a numerical algorithm for a nonlinear transport model

For a nonlinear transport model, we propose a simple and economical two-step algorithm that decreases the dimension of the system of nonlinear equations, as compared with implicit difference schemes. We prove theorems on necessary conditions for stability with respect to the initial data for the nonlinear problem and theorems on sufficient conditions for stability in the case of the linearized model. We also obtain theorems on approximation of the integral conservation law on a grid. The necessary condition obtained is a condition on the coefficients of the differential equation (which singles out an admissible class of equations) but not a condition on the ratio of the grid steps. Bibliography: 3 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 25–32.     

The Nonlinear Initial-Boundary Value Problem and the Existence of Multi-Dimension al Shook Wave for Quasilinear Hyperbolic-Parabolic Coupled Systems

For the quasilinear hyperbolie-parabolio coupled system, the nonlinear initial- boundary value problem and the shook wave free boundary problem are considered. By linear iteration, the existence and uniqueness of the local H^m (m\geq [N+1/2]+4) solution are obtained under the assumption that for the fixed boundary problem, the boundary conditions are uniformly Lopatinski well-posed with respect to the hyperbolic and parabolic part, and for the free boundary problem, there exists a linear stable shock front structure. In particular, the local existence of the isothermal shock wave solution for radiative hydrodynamic eqations is proved.     

New results on the parametric stability of nonlinear systems

In this paper, we derive some new results on the parametric stability of nonlinear systems. Explicitly, we derive a necessary and sufficient condition for a nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. Next, we derive some new results on the parametric stability of discrete-time nonlinear systems. As in the continuous case, we derive a necessary and sufficient condition for a discrete-time nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the discrete-time nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. We illustrate our results with some classical examples from the bifurcation theory.