Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions

In a bounded domain, we consider an Euler–Bernoulli-type thermoelastic plate equation with perturbed boundary conditions. The boundary conditions are such that when the perturbation parameter goes to infinity, we recover the hinged boundary conditions, while one recovers the clamped boundary conditions when the perturbation parameter goes to zero. Relying on resolvent estimates, we show that the underlying semigroup is uniformly, with respect to the perturbation parameter, analytic and exponentially stable. The main features of our proof are appropriate decompositions of the components of the system and the use of Lions? interpolation inequalities.     

On the Spectrum of the Orr–Sommerfeld Equation on the Semiaxis

The Orr–Sommerfeld equation is a spectral problem which is known to play an important role in hydrodynamic stability. For an appropriate operator theoretical realization of the equation, we will determine the essential spectrum, and calculate an enclosure of the set of all eigenvalues by elementary analytical means.     

Rates of decay and growth of solutions to linear stochastic differential equations with state-independent perturbations

The paper studies the almost sure asymptotic convergence to zero of solutions of perturbed linear stochastic differential equations, where the unperturbed equation has an equilibrium at zero, and all solutions of the unperturbed equation tend to zero, almost surely. The perturbation is present in the drift term, and both drift and diffusion coefficients are state‐dependent. We determine necessary and sufficient conditions for the almost sure convergence of solutions to the equilibrium of the unperturbed equation. In particular, a critical polynomial rate of decay of the perturbation is identified, such that solutions of equations in which the perturbation tends to zero more quickly that this rate are almost surely asymptotically stable, while solutions of equations with perturbations decaying more slowly that this critical rate are not asymptotically stable. As a result, the integrability or convergence to zero of the perturbation is not by itself sufficient to guarantee the asymptotic stability of solutions when the stochastic equation with the perturbing term is asymptotically stable. Rates of decay when the perturbation is subexponential are also studied, as well as necessary and sufficient conditions for exponential stability.     

Pointwise Green's Function Estimates Toward Stability for Degenerate Viscous Shock Waves

ABSTRACT

We consider degenerate viscous shock waves arising in systems of two conservation laws, where degeneracy describes viscous shock waves for which the asymptotic endstates are sonic to the hyperbolic system (the shock speed is equal to one of the characteristic speeds). In particular, we develop detailed pointwise estimates on the Green's function associated with the linearized perturbation equation, sufficient for establishing that spectral stability implies nonlinear stability. The analysis of degenerate viscous shock waves involves several new features, such as algebraic (nonintegrable) convection coefficients, loss of analyticity of the Evans function at the leading eigenvalue, and asymptotic time decay of perturbations intermediate between that of the Lax case and that of the undercompressive case.     

Asymptotics of Eigenvalues in the Orr–Sommerfeld Problem for Low Velocities of Unperturbed Flow

Doklady Mathematics - An asymptotic analysis of the eigenvalues and eigenfunctions in the Orr–Sommerfeld problem is carried out in the case when the velocity of the main plane-parallel shear...     

Stabilization of the wave equation with acoustic and delay boundary conditions

In this paper, we consider the wave equation on a bounded domain with mixed Dirichlet-impedance type boundary conditions coupled with oscillators on the Neumann boundary. The system has either a delay in the pressure term of the wave component or the velocity of the oscillator component. Using the velocity as a boundary feedback it is shown that if the delay factor is less than that of the damping factor then the energy of the solutions decays to zero exponentially. The results are based on the energy method, a compactness-uniqueness argument and an appropriate weighted trace estimate. In the critical case where the damping and delay factors are equal, it is shown using variational methods that the energy decays to zero asymptotically.     

Spectral Analysis for Linear Pencils N — λP of Ordinary Differential Operators

Boundary eigenvalue problems for linear pencils N — λ of two ordinary differential operators are studied where P is of lower order than N. In a suitable scale of subspaces of Sobolev spaces and spaces of continuously differentiable functions results on minimality and basis properties of the eigenfunctions and associated functions are proved, including explicit formulas for the Fourier coefficients. As an application the Orr - Sommerfeld equation is considered.     

The stability of a transonic boundary layer on an elastic surface

The perturbations in the boundary layer over an elastic surface when there is non-stationary free viscous-inviscid interaction at transonic velocities are investigated using a modified three-deck model. The modification consists of retaining the term with the second derivative with respect to time (the singular term of the transonic expansion), which occurs in the model of the Lin–Reissner–Tsien equation when it is derived from the complete equations for the velocity potential. This enables the equations of the model to be improved so that they more accurately describe non-stationary and non-linear phenomena. It is shown that the modified model enables perturbations, ignored when using the classical three-deck model, to be taken into account. The compliance on the surface may lead to a reduction in the perturbation growth rate.     

Stability estimates of solutions to extremal problems for a nonlinear convection-diffusion-reaction equation

We consider an identification problem for a stationary nonlinear convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of the substance. This problem is reduced to an inverse extremal problem by an optimization method. The solvability of the boundary value problem and the extremal problem is proved. In the case that the reaction coefficient is quadratic, when the equation acquires cubic nonlinearity, we deduce an optimality system. Analyzing it, we establish some estimates of the local stability of solutions to the extremal problem under small perturbations both of the quality functional and the given velocity vector which occurs multiplicatively in the convection–diffusion–reaction equation.     

Continuous Subsonic-Sonic Flows in a Convergent Nozzle

This paper concerns continuous subsonic-sonic potential flows in a two-dimensional convergent nozzle. It is shown that for a given nozzle which is a perturbation of a straight one, a given point on its wall where the curvature is zero, and a given inlet which is a perturbation of an arc centered at the vertex, there exists uniquely a continuous subsonic-sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve, which satisfies the slip conditions on the nozzle walls and whose sonic curve intersects the upper wall at the given point. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only C^1/2 Hölder continuous and the acceleration blows up. The perturbation problem is solved in the potential plane, where the flow is governed by a free boundary problem of a degenerate elliptic equation with two free boundaries and two nonlocal boundary conditions, and the equation is degenerate at one free boundary.     

Partial stability analysis of some classes of nonlinear systems

A nonlinear differential equation system with nonlinearities of a sector type is studied. Using the Lyapunov direct method and the comparison method, conditions are derived under which the zero solution of the system is stable with respect to all variables and asymptotically stable with respect to a part of variables. Moreover, the impact of nonstationary perturbations with zero mean values on the stability of the zero solution is investigated. In addition, the corresponding time-delay system is considered for which delay-independent partial asymptotic stability conditions are found. Three examples are presented to demonstrate effectiveness of the obtained results.     

Perturbation Sensitivity and Limit Loads of Shells

The perturbation sensitivity and its influence on the limit loads of shells are widely discussed phenomena. Both phenomena may be classified with respect to the type of perturbation. As perturbations influence the stability of shells, the identification of unfavourable perturbations is essential. The perturbation energy concept enables to identify unfavourable non–initial perturbation loads and to evaluate the perturbation sensitivity of fundamental states by the perturbation energy. This measure is also the basis for a load–level–specific optimisation of the perturbation sensitivity. Hence, the present paper discusses primarily the perturbation sensitivity of axially loaded cylindrical shells with different boundary conditions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A version of the asymptotic theory of the unsteady free interaction of a boundary layer with a transonic flow

The non-linear theory of strongly perturbed flows, with a structure of the velocity fields which is characteristic of domains of so-called free interaction of the boundary layer with an external potential flow, is considered. The specific details of the transonic flow show up not only in the estimates of the amplitudes and lengths of the perturbation waves in the asymptotic analysis of the problem but, also, in the fact that the motion may turn out to be simultaneously unsteady in the part of the boundary layer close to the wall and in the external potential flow. This mechanism of the evolution of the perturbations can be described by a single integro-differential equation which, assuming that the structure of the fluctuating fields is a quadrideck structure, is derived using the Fourier-Laplace method. Examples of its non-linear solutions are given in the form of solitary or periodic waves.     

Numerical simulation of seismic activity by the grid-characteristic method

Seismic activity in homogeneous and layered enclosing rock masses is studied. A numerical mechanical-mathematical model of a hypocenter is proposed that describes the whole range of elastic perturbations propagating from the hypocenter. Synthetic beachball plots computed for various fault plane orientations are compared with the analytical solution in the case of homogeneous rock. A detailed analysis of wave patterns and synthetic seismograms is performed to compare seismic activities in homogeneous and layered enclosing rock masses. The influence exerted by individual components of a seismic perturbation on the stability of quarry walls is analyzed. The grid-characteristic method is used on three-dimensional parallelepipedal and curvilinear structured grids with boundary conditions set on the boundaries of the integration domain and with well-defined contact conditions specified in explicit form.     

General Three-Dimensional Disturbances to Inviscid Couette Flow

The general solution to the linearized equations governing three-dimensional disturbances to inviscid Couette flow has been obtained. This result extends the Orr solution to initial conditions that do not consist of a single Fourier sine component in the cross-stream coordinate and a plane wave in the streamwise/spanwise coordinates. The time evolution of a measure of disturbance energy for some specific pulsed initial conditions is examined, and it is concluded that, while the rapid algebraic growth to large amplitude followed by decay exemplified by the Orr solution can be of importance for individual cross-stream Fourier components, more realistic initial conditions, which in general consist of the sum of an infinite number of components, often display uniform decay to zero amplitude. However, an interesting example is described in which one positive definite measure of disturbance amplitude remains constant, yet the streamwise/spanwise velocity components grow linearly in time if the initial disturbance is three-dimensional.     

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.     

一类三阶奇摄动特征值问题

将带有边界条件的三阶非齐次线性方程的可解性条件应用到退化特征值问题上,得出了奇摄动问题的解的渐近表示式.     

The Onset of Linear Instabilities in a Solid Combustion Model

This article concerns the onset of linear instability in a simple model of solid combustion in a semi-infinite two-dimensional strip of width l . The free boundary problem that describes the model involves initial and boundary conditions, including a nonlinear kinetic condition at the interface. The linear problem governing perturbations to a basic solution is solved by the method of images with the reaction front perturbation satisfying an integro-differential equation. This equation is then solved using Laplace transforms. Finally, we perform a stability analysis for the model by studying the solution of the reaction front perturbation. The inclusion of initial conditions enables us to show the development of linear instability from arbitrary initial small disturbances.     

Estimation of vertical wind velocity using doppler radar data with vorticity constraints

The vorticity equation and the mass continuity equation are used as constraints to aid in calculation of the vertical component of a wind field from horizontal wind components. Typically the horizontal wind components result from estimations obtained from radar data. Using a Hilbert space minimization formulation, we characterize the estimated vertical velocity as a solution of an elliptic boundary value problem whose coefficients are functions of the horizontal wind components. Differentiable dependence of the vertical component of the wind field with respect to the horizontal fields is established. Finite dimensional approximating problems are obtained. A numerical study is presented using a sample problem based on the Beltrami flow to compare accuracy of the estimated vertical wind component with the exact. Also, sensitivity to perturbations in horizontal wind components are observed.     

Boundary layer analysis for the eigenvalues of hanging rod

The motion of a naturally straight inextensible flexible elastic hanging rod is formulated and then linearized about the straight solution. To solve this equation by separation of variables, an eigenvalue problem is derived. When the stiffness of the rod is small, the eigenvalue equation is a singular perturbation problem. This paper is devoted to solving this eigenvalue problem by boundary layer analysis when the stiffness is suitably small, especially on the analytic approximate solutions of the first several eigenvalues and eigenfunctions. The first three eigenvalues are also compared with the numerical results computed by a finite difference method. The excellent agreement shows the efficiency of the boundary layer analysis.