Linearized Stability for Semiflows Generated by a Class of Neutral Equations,with Applications to State-Dependent Delays

We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form x′(t) = g(? x _t , x _t ). The state space is a closed subset in a manifold of C ²-functions. Applications include equations with state-dependent delay, as for example x′(t) = a x′(t + d(x(t))) + f (x(t + r(x(t)))) with \({a\in\mathbb{R}, d:\mathbb{R}\to(-h,0), f:\mathbb{R}\to\mathbb{R}, r:\mathbb{R}\to[-h,0]}\).     

Instability of a Longitudinal Movement along the Cylindrical Surface for a Thermoelastic Web Simulated by Stretched and Heated String

On the basis of classical methods for mathematical physics and mechanics, the stability problem of a thermoelastic web moving at a constant speed without friction along a cylindrical surface is investigated. The web is modeled by a stretched and heated string. At a sufficiently high speed and heating of a string, a loss ofmotion stability and the stringmovement in a direction normal to the cylindrical surface occur. To study the instability, a static method based on the consideration of stationary nontrivial modes of stability loss, that is, on the study of the problem for bifurcation of solutions (eigenvalue problem) for the corresponding differential equations is used. The case of the web motion along the circular cylinder is separately considered and an expression for the critical velocity leading to the instability is found.     

Analysis of a Stochastic Implicit Interface Model for an Immersed Elastic Surface in a Fluctuating Fluid

We present some mathematical analyses of a recently proposed stochastic implicit interface model for an elastic surface immersed in an incompressible viscous fluid subject to fluctuation forces. We derive suitable a priori estimates and establish the well-posedness of pathwise solutions and provide uniform control on the solutions in probability.     

关于界面下裂纹对界面裂纹的屏蔽参数

讨论了界面下裂纹对界面裂干涉的合理屏蔽参数问题。对于一条在远场载荷作用下、受到附近界面下裂纹干涉的界面裂纹,采用伪力法计算不同长度单位时界面裂纹的G／G0、k1/k10和k11/k110以及K1／K10和K11／K110（G是能量释放率、k1 ik110是通常定义的应力强度因子、K10＋iK110是含标定长度的应力强度因子;G0、k10 ik110、K10＋iK110对应单一界面裂纹情况）。结果表明,G／G0、K1／K10和K11／K110是较为合理的描述界面下裂纹对界面裂纹屏蔽的参数。     

Relatively Compact Orbits and Compact Attractors for a Class of Nonlinear Evolution Equations

In the present paper we consider the nonlinear evolution equation u+AuG(u), where A:D(A)XX is m-accretive with (I+A)^–1 compact for some >0, and is continuous, and we prove that the orbit is relatively compact if and only if u is uniformly continuous, and both u and G^u are bounded on . In the same spirit, we derive conditions for orbits of bounded sets to have compact attractors. Some consequences and an example from age-structured population dynamics illustrate the effectiveness of the abstract result.     

New Explicit and Exact Travelling Wave Solutions for a Class of Nonlinear Evolution Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｔｈｅｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｏｆｓｃｉｅｎｃｅａｎｄｔｅｃｈｎｏｌｏｇｙ ,ｔｈｅｓｔｕｄｙｋｅｒｎｅｌｏｆｍｏｄｅｒｎｓｃｉｅｎｃｅｉｓｃｈａｎｇｅｄｆｒｏｍｌｉｎｅａｒｔｏｎｏｎｌｉｎｅａｒｓｔｅｐｂｙｓｔｅｐ .Ｍａｎｙｎｏｎｌｉｎｅａｒｓｃｉｅｎｃｅｐｒｏｂｌｅｍｓｃａｎｓｉｍｐｌｙａｎｄｅｘａｃｔｌｙｂｅｄｅｓｃｒｉｂｅｄｂｙｕｓｉｎｇｔｈｅｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌｏｆｎｏｎｌｉｎｅａｒｅｑｕａｔｉｏｎ .Ｕｐｔｏｎｏｗ ,ｍａｎｙｉｍｐｏｒ…     

Carleman Estimate for Elliptic Operators with Coefficients with Jumps at an Interface in Arbitrary Dimension and Application to the Null Controllability of Linear Parabolic Equations

In a bounded domain of R ⁿ⁺¹, n ≧ 2, we consider a second-order elliptic operator, ${A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}In a bounded domain of R ⁿ⁺¹, n ≧ 2, we consider a second-order elliptic operator, A=-?_x0² - ?_x ·(c(x) ?_x){A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}, where the (scalar) coefficient c(x) is piecewise smooth yet discontinuous across a smooth interface S. We prove a local Carleman estimate for A in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient c at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions and the Calderón projector technique. Following the method of Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) we then prove the null controllability for the linear parabolic initial problem with Dirichlet boundary conditions associated with the operator ?_t - ?_x ·(c(x) ?_x){{\partial_t - \nabla_x \cdot (c(x) \nabla_x)}} .     

Estimating the Lyapunov Function and Stability of Motion of a System with Equations of Motion with an Asymptotically Expanded Right-Hand Side

International Applied Mechanics - New estimates of the Lyapunov function along the solutions of a system with an asymptotically expanded right-hand side are established. The deviation of the...     

Bulk Cavitation and the Possibility of Localized Interface Deformation due to Surface Layer Swelling

We consider the uniform swelling of a compressible hyperelastic surface layer with finite thickness that is attached to an underlying bulk material composed of a non-swelling incompressible hyperelastic material. In addition to classically smooth solutions, two additional phenomena may occur for sufficiently large swelling. One is the formation of cavities in the interior of the underlying bulk material. The other is the disappearance of smooth solutions in the surface layer while the underlying bulk material remains intact. It is conjectured that the latter may be associated with the concentration of deformation at the swelling interface. Both phenomena are investigated by the consideration of solutions to a boundary value problem for a sphere involving radial deformation with a prescribed swelling field that acts as an effective loading device. Specific material models for both the compressible swollen surface layer and the non-swollen incompressible bulk are invoked so as to permit an analytical treatment. Swelling thresholds are obtained that depend on the thickness of the surface layer for the onset of these separate phenomena.     

Stability and Stress State of an Underground Pipeline Joined to a Unit

International Applied Mechanics - The stability of a pipeline jointed to a unit interacting with the surrounding soil is analyzed. The pipeline is subject to axial compression. The stress state of...     

New Transforms for the Resolving Equations in Elastic Theory and New Integral Transforms, with Applications to Boundary-Value Problems of Mechanics

This paper reviews results from an analysis of exact boundary-value solutions of static and dynamic elasticity obtained by the method of integral transforms. Consideration is given to the solutions of problems for a half-space and a blunted hollow cone, the heat conduction problem for a cone, and the diffraction problem for a cone with a hole along the generatrix. Solutions of mechanics problems are analyzed and the method of integral transforms is generalized     

About the Linear Stability of the Spherically Symmetric Solution for the Equations of a Barotropic Viscous Fluid under the Influence of Self-Gravitation

This paper is concerned with the question of linear stability of motionless, spherically symmetric equilibrium states of viscous, barotropic, self-gravitating fluids. We prove the linear asymptotic stability of such equilibria with respect to perturbations which leave the angular momentum, momentum, mass and the position of the center of gravity unchanged. We also give some decay estimates for such perturbations, which we derive from resolvent estimates by means of analytic semigroup theory.     

Regularity of Solutions to the Navier-Stokes Equations for Compressible Barotropic Flows on a Polygon

The Navier-Stokes system for a steady-state barotropic nonlinear compressible viscous flow, with an inflow boundary condition, is studied on a polygon D. A unique existence for the solution of the system is established. It is shown that the lowest order corner singularity of the nonlinear system is the same as that of the Laplacian in suitable L ^q spaces. Let ω be the interior angle of a vertex P of D. If \(\) and \(\), then the velocity u is split into singular and regular parts near the vertex P. If α < 2 and \(\) or if α > 2 and 2 < q < ∞&;, it is shown that u∈ (H ^{2, q} (D))².     

Flow and Mass Transfer in the Presence of an Unsteady Supply of Solution to the Surface of a Growing Crystal

Mixed convection in a highly supersaturated isothermal aqueous solution of potassium dihydrophosphate at the surface of a growing single crystal is investigated numerically. Unsteady solution flow at the feeder exit is simulated by specifying a time-periodic velocity distribution. It is shown that the concentration boundary layer is significantly homogenized as compared with a steady-state supply. The nature of the flow is investigated as a function of the distance between the feeder and the crystal surface.     

求解饱和半空间上弹性圆板固结沉降的积分方程

本文采用解析方法分析了弹性圆板在饮和半空间上的固结沉降。考虑弹性圆板与饮和半空间的接触面上无摩擦力,且饱和半空间表面为全部透水的。运用Ｂｉｏｔ固结理论和积分方程技术,在Ｌａｐｌａｃｅ变换域上建立了弹性圆板固结沉降的对偶积分方程,并化此对偶积分方程为第二类Ｆｒｅｄｈｏｌｍ积分方程。通过对其核函数的有效数值发得到第二类Ｆｒｅｄｈｏｌｍ积分方程的解,再利用Ｌａｐａｃｅ反演技术获得弹性板在时间域中的固结沉     

Generalization of Vlasov’s Equations for a Cylindrical Shell to the Case of a Transversely Isotropic Material

Differential equations of the general theory of transversely isotropic cylindrical shells are obtained; in a certain sense, these equations are generalizations of Vlasov’s and Ambartsumyan’s equations. This allowed us on the basis of Novozhilov’s criterion (comparison of variability of the stress state in the principal orthogonal directions) to divide the initial equations according to Goldenweiser into approximate equations of the type of the semi-momentless theory, theory of the edge effect and flexural state, which are also generalizations of equations that describe the elementary stress states of an isotropic shell. Numerical values are found for criteria of matching of approximate equations that describe the elementary stress states in the asymptotic synthesis of the full stress state. Examples of calculations and experimental data for a shell with and without allowance for transverse shear strain are given.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 125–132, July– August, 2005.     

Stability of a volumetrically charged dielectric fluid jet accelerated in an electric field collinear to the jet

In the approximation linear in the amplitude of capillary waves (generally nonaxisymmetric) on the surface of a volumetrically charged dielectric fluid jet accelerated in an external electrostatic field collinear to the jet axis, a dispersion equation is derived and analyzed. It is shown that for certain values of physical parameters the effect of the external electric field stabilizing the capillary waves and the destabilizing effect of the capillary forces and the electrostatic forces exerted by the jet charge may compensate one another for both axisymmetric and nonaxisymmetric, with a high degree of asymmetry, waves. The instability of flexural waves cannot be suppressed completely and can be realized for sufficiently small wave numbers.