On the Large Time Behavior of Solutions of Hamilton–Jacobi Equations Associated with Nonlinear Boundary Conditions

In this article, we study the large time behavior of solutions of first-order Hamilton–Jacobi Equations set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy–Neumann problems by using two fairly different methods: the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the “weak KAM approach”, which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry–Mather sets.     

Propagation of Density-Oscillations in Solutions to the Barotropic Compressible Navier–Stokes System

Considering a bounded sequence of weak solutions to the compressible Navier–Stokes system, we introduce Young measures as in [12] in order to describe a “homogenized system” satisfied in the limit. We then study the Cauchy problem associated to this “homogenized system” when Young measures are convex combinations of Dirac measures.     

On the extendability of solutions of a difference equation “to the left”

We give sufficient conditions for the extendability of solutions of a nonlinear difference equation “to the left” in a Banach space. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 3, pp. 298–302, July–September, 2007.     

Asymptotics of solutions of the three-dimensional elasticity equations for compressible and incompressible bodies

We analyze the leading terms in the general asymptotic expansions of solutions of the first boundary value problem of three-dimensional elasticity in displacements. The cases of compressible and incompressible bodies, which have substantially different statements, are considered separately. The minimum-to-maximum ratio of characteristic dimensions of the elastic body is a natural small asymptotic parameter. The third dimension can be of any “intermediate” order, including the endpoints. For example, such a geometry is typical of bodies that simultaneously have characteristic macro-, micro-, and nano-dimensions in three coordinate axes.     

Stability of equilibrium for an inverted two-link mathematical pendulum with critical tracking forces

We investigate nonlinear stability for equilibrium of a pendulum with viscoelastic components. The tracking force is chosen so that the matrix of the linearized part of the perturbed motion has two purely imaginary roots or one zero and one negative root. The other two roots are complex with negative real part. The boundary of the domain of stability is divided into “dangerous” and “safe” (in the sense of Bautin) zones. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 100–105, September, 1999.     

Construction of asymptotic solutions of a singularly perturbed linear system of differential equations with irregular singular point in the case of a multiple spectrum of the boundary matrix

We construct asymptotic solutions of a singularly perturbed linear system of differential equations with irregular singular point. We consider the case where the main matrix has a multiple eigenvalue associated with one or several elementary divisors of the same multiplicity. We establish that, in the case of multiple elementary divisors, the corresponding asymptotic expansions can be constructed in the form of double series in fractional powers of the parameter and the ratio of the independent variable to the parameter. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 1, pp. 128–144, January–March, 2007.     

On Lyapunov stability of impulsive Takagi–Sugeno fuzzy systems

We analyze the Lyapunov stability of impulsive Takagi–Sugeno fuzzy systems. Using the direct Lyapunov method, we establish sufficient conditions for the stability of these systems. We show that these conditions can be expressed in terms of a system of linear matrix inequalities. As an example, we consider an impulsive fuzzy control in a two-species “predator–prey” model. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 481–494, October–December, 2008.     

Initial layer phenomena for a class of singular perturbed nonlinear system with slow variables

The initial layer phenomena for a class of singular perturbed nonlinear system with slow variables are studied. By introducing stretchy variables with different quantity levels and constructing the correction term of initial layer with different “ thickness“, the Norder approximate expansion of perturbed solution concerning small parameter is obtained, and the “ multiple layer“ phenomena of perturbed solutions are revealed. Using the fixed point theorem, the existence of perturbed solution is proved, and the uniformly valid asymptotic expansion of the solutions is given as well.     

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULA RPLATE（I）

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Addendum to: Symmetry classification of quasi-linear PDE’s. II: an exceptional case

This short note completes the symmetry analysis of a class of quasi-linear partial differential equations considered in the previous paper (Nonlinear Dynamics 51: 309–316, 2008): it deals with an “exceptional” Lie point symmetry which is admitted only if the involved parameters are fixed by precise relationships. The peculiarity of this symmetry is enhanced by the fact that, combined with the presence of a conditional symmetry of “weak” type, it leads to a family of solutions which include, as a particular case, a relevant solution of the Grad–Schlüter–Shafranov equation, well known in plasma physics.     

Development of an X-ray computed tomography (CT) system with sparse sources: application to three-phase pipe flow visualization

This paper describes the application of a two-beam X-ray computed tomography (CT) system to multiphase (gas–oil–water) flow measurement. Two high-voltage (160 keV) X-ray sources are used to penetrate a 4-in. (101.6 mm ID) pipeline. A rotating filter wheel mechanism is employed to alternately “harden” and “soften” the X-ray spectra to provide discrimination between the three phases. Because this system offers only two projections, conventional back-projection algorithms are ineffective and thus a new reconstruction technique has been developed. A matrix equation is formed, to which additional “smoothing equations” are added to compensate for the lack of projection data. The tomographic result is obtained by computing an inverse matrix. This is a one-off computation and the inverse is stored for repeated use; reconstructed images from synthesized data demonstrate the effectiveness of this technique. Three-phase tomographic images of a horizontal slug flow are presented, which clearly show the mixing of oil and water layers within the slug body. The relevance of this work to the offshore oil and gas industry is summarized.     

Qualitative Behavior of Solutions of a Randomly Perturbed Reaction-Diffusion Equation

On the basis of the developed abstract theory of random attractors of probability dissipative systems, we investigate the qualitative behavior of solutions of a nonuniquely solvable reaction-diffusion equation perturbed by a stochastic “cadlag” process. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 174–185, April–June, 2005.     

Heat transfer characteristics of the algebraically decaying Glauert jet

The classical exponentially decaying wall jet considered independently by Tetervin (NACA TN 1644 40 pp, 1948), Akatnov (Leningrad Politek Inst Trudy 5:24–31, 1953) and Glauert (J Fluid Mech 1:625–643, 1956) as well as its algebraically decaying counterpart (which will be referred to hereafter as “algebraic Glauert Jet”, or AG-jet for short) belong to the same similarity class of solutions of the boundary layer equations. We investigate in this paper the thermal characteristics of a nonpreheated AG-jet over a permeable wall for prescribed constant wall temperature and prescribed constant heat flux. Their scaling behavior for small and large values of the Prandtl number is discussed in detail and compared to that of the classical Tetervin–Akatnov–Glauert wall jet.     

Nonlinearly Elastic Shell Models: A Formal Asymptotic Approach I. The Membrane Model

We consider a family of three‐dimensional shells with the same middle surface, all composed of the same nonlinearly elastic Saint Venant‐Kirchhoff material. Using the method of asymptotic expansions with the thickness as the “small” parameter, and making specific assumptions on the applied forces, the geometry of the middle surface, and the kinematic boundary conditions, we show how a “limiting”, “large‐deformation” two‐dimensional model can be identified in this fashion. By linearization, this nonlinear membrane model reduces to the linear membrane model. (Accepted January 13, 1997)     

Effects of the weak/micro-discontinuity of interface on the fracture behavior of a functionally graded coating with an inclined crack

The mechanical model was established for the anti-plane fracture problem of a functionally graded coating–substrate system with a coating crack inclined to the weak/micro-discontinuous interface. The Cauchy singular integral equation for the crack was derived using Fourier integral transform, and the Lobatto–Chebyshev collocation method put up by Erdogan and Gupta was used to get its numerical solution. Finally, the effects of the weak/micro-discontinuity of the interface on SIFs were analyzed, the “affected regions” corresponding to the two crack tips have been obtained and their engineering significance was discussed. It was indicated that, for the crack tip in the corresponding “affected region”, to reduce the weak-discontinuity of the interface and to make the interface micro-discontinuous are the two effective ways to reduce the SIF, and the latter way always has more remarkable SIF-reduction effect. For the crack tip outside the “affected region”, its SIF is mainly influenced by material stiffness, and to prevent such a tip from growing toward the interface “softer coating and stiffer substrate” is a more advantageous combination than “stiffer coating and softer substrate”.     

The thomson friction self-oscillations of a two-mass system with delay

A technique is proposed to study and design a mechanical self-oscillating system in the quasiharmonic-oscillation regime. The technique is based on the polynomial approximation of the force due to dry sliding friction by a finite number of terms in the Taylor-series expansion with allowance for energy dissipation in accordance with Pisarenko's hypothesis and the first “improved” asymptotic approximation. Transport University, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 36, No. 7, pp. 137–144, July, 2000.     

Formation of Symmetric Structures in the Dynamics of Repelling Particles

We consider Hamiltonian systems corresponding to the motions of a system of N repelling particles evolving in space under the action deriving from a very long range potential energy; the asymptotic behavior of the system is analysed for the cases U=− ln r and . Only special “asymptotic?shapes” are reached, which may present quite interesting symmetries and correspond to the critical points of a gradient system. The relationships between the original Hamiltonian and the asymptotic gradient system are discussed. Accepted: May 25, 1999     

Soft and Hard Wall in a Stochastic Reaction Diffusion Equation

We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prove that, in a suitable scaling limit, the front evolves according to a one-dimensional diffusion process with a non-linear drift accounting for a “soft” repulsion from the boundary. We finally show how a “hard” repulsion can be obtained by an extra diffusive scaling.     

Smoothing Estimates for Boltzmann Equation with Full-Range Interactions: Spatially Homogeneous Case

In this work, we are concerned with the regularities of the solutions to the Boltzmann equation with physical collision kernels for the full range of intermolecular repulsive potentials, r ^−(p−1) with p > 2. We give new and constructive upper and lower bounds for the collision operator in terms of standard weighted fractional Sobolev norms. As an application, we get the global entropy dissipation estimate which is a little stronger than that described by Alexandre et al. (Arch Rational Mech Anal 152(4):327–355, 2000). As another application, we prove the smoothing effects for the strong solutions constructed by Desvillettes and Mouhot (Arch Rational Mech Anal 193(2):227–253, 2009) of the spatially homogeneous Boltzmann equation with “true” hard potential and “true” moderately soft potential.