Projection-algebraic approximation of linear and nonlinear operator differential equations in Banach spaces

We develop a Calogero-type projection-algebraic method of discrete approximations for linear differential equations in Banach spaces and analyze the convergence of finite-dimensional approximations based on the functional-analytic approach to discrete approximations and methods of operator theory in Banach spaces. Applications of the obtained results to the functional-interpolation scheme of the projection-algebraic method of discrete approximations are considered. Based on a generalized Leray–Schauder-type theorem, we consider the projection-algebraic scheme of discrete approximations and analyze its solvability and convergence for a special class of nonlinear operator equations.     

Invariant sets and comparison of dynamical systems

We propose a method for the construction and investigation of invariant sets of differential systems described by cone inequalities with the use of the operator of differentiation along the trajectories of the system. Well-known conditions for the positivity of linear and nonlinear differential systems with respect to typical classes of cones are generalized. A method for comparison and ordering is developed for a family of dynamical systems. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 2, pp. 163–176, April–June, 2007.     

Fokker–Planck Equations for a Free Energy Functional or Markov Process on a Graph

The classical Fokker–Planck equation is a linear parabolic equation which describes the time evolution of the probability distribution of a stochastic process defined on a Euclidean space. Corresponding to a stochastic process, there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and an entropy. In recent years, it has been shown that the Fokker–Planck equation is the gradient flow of the free energy functional defined on the Riemannian manifold of probability distributions whose inner product is generated by a 2-Wasserstein distance. In this paper, we consider analogous matters for a free energy functional or Markov process defined on a graph with a finite number of vertices and edges. If N ≧ 2 is the number of vertices of the graph, we show that the corresponding Fokker–Planck equation is a system of N nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. However, in contrast to stochastic processes defined on Euclidean spaces, the situation is more subtle for discrete spaces. We have different choices for inner products on the space of probability distributions resulting in different Fokker–Planck equations for the same process. It is shown that there is a strong connection but there are also substantial discrepancies between the systems of ordinary differential equations and the classical Fokker–Planck equation on Euclidean spaces. Furthermore, both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the Riemannian manifolds of probability distributions with different metrics. Some examples are also discussed.     

Averaging of initial-value and boundary-value problems for one class of oscillatory impulsive systems

We give a new classification of fixed times of pulse action (uniform, functional, limiting, and quantitatively limiting). Several results obtained earlier for oscillatory systems with uniform and limiting times of pulse action are generalized to similar systems with functional times of pulse action. Namely, we obtain estimates, which are exact with respect to a small parameter ε, for the deviation of solutions and their partial derivatives for original and averaged initial-value, boundary-value, and multipoint problems. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 68–84, January–March, 2006.     

Fundamental Solutions of the Cauchy Problem for Invariant Λ<Subscript>(µ)</Subscript>-Hyperbolic Operators on Riemannian Manifolds

We obtain an analytic representation of a fundamental solution of the Cauchy problem for Petrovskii strictly hyperbolic Λ_(μ)-invariant hyperbolic equations and systems of equations in Euclidean spaces and on special Riemann manifolds on the basis of the introduced integral transformations generated by an integral representation of the Dirac measure. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 224–233, April–June, 2005.     

Hamiltonian finite-dimensional oscillator-type reductions of Lax-integrable superconformal hierarchies

For bi-Hamiltonian superconformal hierarchies of nonlinear Benny-Kaup and Kaup-Broer dynamical systems, we develop a method for reduction to nonlocal finite-dimensional invariant subspaces of Neumann and Bargmann types, respectively. We prove that there exist even supersymplectic structures on these spaces and that the reduced commuting vector fields generated by the hierarchies are integrable in the Lax-Liouville sense. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 15–30, January–March, 2006.     

A Comparison Principle for Hamilton–Jacobi Equations Related to Controlled Gradient Flows in Infinite Dimensions

We develop new comparison principles for viscosity solutions of Hamilton–Jacobi equations associated with controlled gradient flows in function spaces as well as the space of probability measures. Our examples are optimal control of Ginzburg–Landau and Fokker–Planck equations. They arise in limit considerations of externally forced non-equilibrium statistical mechanics models, or through the large deviation principle for interacting particle systems. Our approach is based on two key ingredients: an appropriate choice of geometric structure defining the gradient flow, and a free energy inequality resulting from such gradient flow structure. The approach allows us to handle Hamiltonians with singular state dependency in the nonlinear term, as well as Hamiltonians with a state space which does not satisfy the Radon–Nikodym property. In the case where the state space is a Hilbert space, the method simplifies existing theories by avoiding the perturbed optimization principle.     

Bounded solutions of linear differential equations in a banach space

For a linear inhomogeneous differential equation in a Banach space, we find a criterion for the existence of solutions that are bounded on the entire real axis under the assumption that the homogeneous equation admits an exponential dichotomy on the semiaxes. This result is a generalization of the Palmer lemma to the case of infinite-dimensional spaces. We consider examples of countable systems of ordinary differential equations that have bounded solutions. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 3–14, January–March, 2006.     

Stability for the 3D Navier–Stokes Equations with Nonzero far Field Velocity on Exterior Domains

Concerning to the non-stationary Navier–Stokes flow with a nonzero constant velocity at infinity, just a few results have been obtained, while most of the results are for the flow with the zero velocity at infinity. The temporal stability of stationary solutions for the Navier–Stokes flow with a nonzero constant velocity at infinity has been studied by Enomoto and Shibata (J Math Fluid Mech 7:339–367, 2005), in L ^p spaces for p ≥ 3. In this article, we first extend their result to the case \frac32 < p{\frac{3}{2} < p} by modifying the method in Bae and Jin (J Math Fluid Mech 10:423–433, 2008) that was used to obtain weighted estimates for the Navier–Stokes flow with the zero velocity at infinity. Then, by using our generalized temporal estimates we obtain the weighted stability of stationary solutions for the Navier–Stokes flow with a nonzero velocity at infinity.     

Optimal nonlinear observers for chaotic synchronization with message embedded

This paper presents an optimal nonlinear observer for synchronizing the transmitter-receiver pair with guaranteed optimal performance. In the proposed scheme, a generalized nonlinear state-space observer via uniform matrix transformations is constructed to estimate the transmitter state and the information signal, simultaneously. A nonlinear optimal design approach is used to synchronize chaotic systems. Solving the Hamilton–Jacobi–Bellman (H–J–B) equations we can obtain a linear optimal feedback scheme for piecewise-linear chaotic systems. Moreover, a robust scheme derived from the H _∞ optimization theory improves the synchronization performance of general nonlinear chaotic systems by suppressing the influence of their high order residual terms. Finally, two numerical simulation examples are illustrated by the chaotic Chua’s circuit system and the Lorenz chaotic system to demonstrate the effectiveness of our scheme.     

Analyticity and Decay Estimates of the Navier–Stokes Equations in Critical Besov Spaces

In this paper, we establish analyticity of the Navier–Stokes equations with small data in critical Besov spaces . The main method is Gevrey estimates, the choice of which is motivated by the work of Foias and Temam (Contemp Math 208:151–180, 1997). We show that mild solutions are Gevrey regular, that is, the energy bound holds in , globally in time for p < ∞. We extend these results for the intricate limiting case p = ∞ in a suitably designed E _∞ space. As a consequence of analyticity, we obtain decay estimates of weak solutions in Besov spaces. Finally, we provide a regularity criterion in Besov spaces.     

Numerical simulation of nonholonomic rigid-body systems

The paper proposes computer algebra system (CAS) algorithms for computer-assisted derivation of the equations of motion for systems of rigid bodies with holonomic and nonholonomic constraints that are linear with respect to the generalized velocities. The main advantages of using the D’Alembert-Lagrange principle for the CSA-based derivation of the equations of motion for nonholonomic systems of rigid bodies are demonstrated. Among them are universality, algorithmizability, computational efficiency, and simplicity of deriving equations for holonomic and nonholonomic systems in terms of generalized coordinates or pseudo-velocities __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 9, pp. 106–115, September 2006.     

Estimating the Largest Lyapunov Exponent in a Multibody System with Dry Friction by using Chaos Synchronization

Using the properties of chaos synchronization, the method for estimating the largest Lyapunov exponent in a multibody system with dry friction is presented in this paper. The Lagrange equations with multipliers of the systems are given in matrix form, which is adequate for numerical calculation. The approach for calculating the generalized velocity and acceleration of the slider is given to determine slipping or sticking of the slider in the systems. For slip–slip and stick–slip multibody systems, their largest Lyapunov exponents are calculated to characterize their dynamics.The project supported by the National Natural Science Foundation of China (10272008 and 10371030) The English text was polished by Yunming Chen     

Prevalent Behavior of Strongly Order Preserving Semiflows

Classical results in the theory of monotone semiflows give sufficient conditions for the generic solution to converge toward an equilibrium or toward the set of equilibria (quasiconvergence). In this paper, we provide new formulations of these results in terms of the measure-theoretic notion of prevalence, developed in Christensen (Israel J. Math., 13, 255–260, 1972) and Hunt et al. (Bull. Am. Math. Soc., 27, 217–238, 1992). For monotone reaction–diffusion systems with Neumann boundary conditions on convex domains, we show the prevalence of the set of continuous initial conditions corresponding to solutions that converge to a spatially homogeneous equilibrium. We also extend a previous generic convergence result to allow its use on Sobolev spaces. Careful attention is given to the measurability of the various sets involved.     

A Semi-Analytical Solution for Large-Scale Injection-Induced Pressure Perturbation and Leakage in a Laterally Bounded Aquifer–Aquitard System

A number of (semi-)analytical solutions are available to drawdown analysis and leakage estimation of shallow aquifer–aquitard systems. These solutions assume that the systems are laterally infinite. When a large-scale pumping from (or injection into) an aquifer–aquitard system of lower specific storativity occurs, induced pressure perturbation (or hydraulic head drawdown/rise) may reach the lateral boundary of the aquifer. We developed semi-analytical solutions to address the induced pressure perturbation and vertical leakage in a “laterally bounded” system consisting of an aquifer and an overlying/underlying aquitard. A one-dimensional radial flow equation for the aquifer was coupled with a one-dimensional vertical flow equation for the aquitard, with a no-flow condition imposed on the outer radial boundary. Analytical solutions were obtained for (1) the Laplace-transform hydraulic head drawdown/rise in the aquifer and in the aquitard, (2) the Laplace-transform rate and volume of leakage through the aquifer–aquitard interface integrated up to an arbitrary radial distance, (3) the transformed total leakage rate and volume for the entire interface, and (4) the transformed horizontal flux at any radius. The total leakage rate and volume depend only on the hydrogeologic properties and thicknesses of the aquifer and aquitard, as well as the duration of pumping or injection. It was proven that the total leakage rate and volume are independent of the aquifer’s radial extent and wellbore radius. The derived analytical solutions for bounded systems are the generalized solutions of infinite systems. Laplace-transform solutions were numerically inverted to obtain the hydraulic head drawdown/rise, leakage rate, leakage volume, and horizontal flux for given hydrogeologic and geometric conditions of the aquifer–aquitard system, as well as injection/pumping scenarios. Application to a large-scale injection-and-storage problem in a bounded system was demonstrated.     

Invariant Measures for Dissipative Systems and Generalised Banach Limits

Inspired by a theory due to Foias and coworkers (see, for example, Foias et al. Navier–Stokes equations and turbulence, Cambridge University Press, Cambridge, 2001) and recent work of Wang (Disc Cont Dyn Sys 23:521–540, 2009), we show that the generalised Banach limit can be used to construct invariant measures for continuous dynamical systems on metric spaces that have compact attracting sets, taking limits evaluated along individual trajectories. We also show that if the space is a reflexive separable Banach space, or if the dynamical system has a compact absorbing set, then rather than taking limits evaluated along individual trajectories, we can take an ensemble of initial conditions: the generalised Banach limit can be used to construct an invariant measure based on an arbitrary initial probability measure, and any invariant measure can be obtained in this way. We thus propose an alternative to the classical Krylov–Bogoliubov construction, which we show is also applicable in this situation.     

A generalized algebraic method of new explicit and exact solutions of the nonlinear dispersive generalized Benjiamin–Bona–Mahony equations

Nonlinear dispersive generalized Benjiamin–Bona–Mahony equations are studied by using a generalized algebraic method. New abundant families of explicit and exact traveling wave solutions, including triangular periodic, solitary wave, periodic-like, soliton-like, rational and exponential solutions are constructed, which are in agreement with the results reported in other literatures, and some new results are obtained. These solutions will be helpful to the further study of the physical meaning and laws of motion of the nature and the realistic models. The proposed method in this paper can be further extended to the 2+1 dimensional and higher dimensional nonlinear evolution equations or systems of equations.     

On the Amplitude Equations for Weakly Nonlinear Surface Waves

Nonlocal generalizations of Burgers’ equation were derived in earlier work by Hunter (Contemp Math, vol 100, pp 185–202. AMS, 1989), and more recently by Benzoni-Gavage and Rosini (Comput Math Appl 57(3–4):1463–1484, 2009), as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage (Differ Integr Equ 22(3–4):303–320, 2009) under an appropriate stability condition originally pointed out by Hunter. The same stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov (Benzoni-Gavage et al. in Adv Math 227(6):2220–2240, 2011). In this article, we show how the verification of Hunter’s stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. We also show that the resulting amplitude equation has a Hamiltonian structure when the original boundary value problem has a variational origin. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity.     

Generalized green operator for an impulsive boundary-value problem with switchings

We establish constructive existence conditions and construct a generalized Green operator for the construction of solutions of a Noetherian linear boundary-value problem for a system of ordinary differential equations with switchings and pulse action in critical and noncritical cases. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 1, pp. 51–65, January–March, 2007.     

Regular integral equations for the second boundary-value problem of the bending of an anisotropic elastic plate

Two systems of Fredholm equations of the second kind are constructed for the solution of the second boundary-value problem of the bending of an anisotropic plate (a normal bending moment and a generalized shear force are specified on the boundary of the simply-connected domain) under the assumption of validity of the Kirchhoff-Love hypotheses. Correct equilibrium conditions are specified for the examined boundary-value problem. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 108–119, May–June, 2005.