Derivation of equations of seismic and acoustic wave propagation and equations of filtration via homogenization of periodic structures

A linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space is considered. Although the problem is linear, it is very hard to tackle due to the fact that its main differential equations involve nonsmooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As the results for different ratios between physical parameters, we derive Biot’s equations of poroelasticity, a system consisting of nonisotropic Lamé’s equations for the solid component and acoustic equations for the liquid component, nonisotropic Lamé’s equations or equations of viscoelasticity for one-velocity continuum, decoupled system consisting of Darcy’s system of filtration or acoustic equations for the liquid component (first approximation) and nonisotropic Lamé’s equations for the solid component (second approximation), a system consisting of nonisotropic Stokes equations for the liquid component and acoustic equations for the solid component, nonisotropic Stokes equations for one-velocity continuum, or, finally a different type of acoustic equations for one- or two-velocity continuum. The proofs are based on Nguetseng’s two-scale convergence method of homogenization in periodic structures.     

The general equations of analytical dynamics

It is shown that the generalized Poincaré and Chetayev equations, which represent the equations of motion of mechanical systems using a certain closed system of infinitesimal linear operators, are related to the fundamental equations of analytical dynamics. Equations are derived in quasi-coordinates for the case of redundant variables; it is shown that when an energy integral exists the operator X₀ = ∂/∂t satisfies the Chetayev cyclic-displacement conditions. Using the energy integral the order of the system of equations of motion is reduced, and generalized Jacobi-Whittaker equations are derived from the Chetayev equations. It is shown that the Poincaré-Chetayev equations are equivalent to a number of equations of motion of non-holonomic systems, in particular, the Maggi, Volterra, Kane, and so on, equations. On the basis of these, and also of other previously obtained results, the Poincaré and Chetayev equations in redundant variables, applicable both to holonomic and non-holonomic systems, can be regarded as general equations of classical dynamics, equivalent to the well-known fundamental forms of the equations of motion, a number of which follow as special cases from the Poincaré and Chetayev equations.     

Existence of solutions for the MHD-Leray-alpha equations and their relations to the MHD equations

In this paper we derive some new equations and we call them MHD-Leray-alpha equations which are similar to the MHD equations. We put forward the concept of weak and strong solutions for the new equations. Whether the 3-dimensional MHD equations have a unique weak solution is unknown, however, there is a unique weak solution for the 3-dimensional MHD-Leray-alpha equations. The global existence of strong solution and the Gevrey class regularity for the new equations are also obtained. Furthermore, we prove that the solutions of the MHD-Leray-alpha equations converge to the solution of the MHD equations in the weak sense as the parameter ε in the new equations converges to zero.     

Structure of the set of stationary solutions of phase-lock equations in superconductivity

In previous article [M. Zhan, Phase-lock equations and its connections to Ginzburg–Landau equations of superconductivity, J. Nonlinear Anal. 42 (2000) 1063–1075], we introduced a system of equations (phase-lock equations) to model the superconductivity phenomena. We investigated its connection to Ginzburg–Landau equations and proved the existence and uniqueness of both weak and strong solutions. In this article, we study the steady-state problem associated with the phase-lock equations. We prove that the steady-state problem has multiple solutions and show that the solution set enjoys some structural properties as proved by Foias and Teman for the Navier–Stokes equations in [C. Foias, R. Teman, Structure of the set of stationary solutions of the Navier–Stokes equations, Commun. Pure Appl. Math. XXX (1977) 149–164].     

One method for solving systems of nonlinear partial differential equations

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).     

On the Nevanlinna Order of Meromorphic Solutions to Linear Analytic Difference Equations

For various classes of linear ordinary analytic difference equations with meromorphic coefficients, we study Nevanlinna order properties of suitable meromorphic solutions. For a large class of first-order equations with coefficient of order ρ∈[0, ∞), we explicitly construct meromorphic solutions of order ≤ρ+ 1. For higher-order equations with coefficients of order ρ∈[0, ∞), we show that meromorphic solutions with increase of order ≤ρ+ 1 in a certain strip have order ≤ρ+ 1. The assumptions made in the latter setting may seem quite restrictive, but they are satisfied for several classes of second-order difference equations that have been studied in recent years. The latter include Harper-type equations, "reflectionless" equations, Askey–Wilson-type equations, and equations of relativistic Calogero–Moser type.     

On the shape of solution sets of systems of (functional) equations

Solution sets of systems of linear equations over fields are characterized as being affine subspaces. But what can we say about the “shape” of the set of all solutions of other systems of equations? We study solution sets over arbitrary algebraic structures, and we give a necessary condition for a set of n-tuples to be the set of solutions of a system of equations in n unknowns over a given algebra. In the case of Boolean equations we obtain a complete characterization, and we also characterize solution sets of systems of Boolean functional equations.     

The macroscopic system of Einstein-Maxwell equations for a system of interacting particles

We derive the macroscopic Einstein—Maxwell equations up to the second-order terms, in the interaction for systems with dominating electromagnetic interactions between particles (e.g., radiation-dominated cosmological plasma in the expanding Universe before the recombination moment). The ensemble averaging of the microscopic Einstein and Maxwell equations and of the Liouville equations for the random functions of each type of particle leads to a closed system of equations consisting of the macroscopic Einstein and Maxwell equations and the kinetic equations for one-particle distribution functions for each type of particle. The macroscopic Einstein equations for a system of electromagnetically and gravitationally interacting particles differ from the classical Einstein equations in having additional terms in the lefthand side due to the interaction. These terms are given by a symmetric rank-two traceless tensor with zero divergence. Explicitly, these terms are represented as momentum-space integrals of the expressions containing one-particle distribution functions for each type of particle and have much in common with similar terms in the left-hand side of the macroscopic Einstein equations previously obtained for a system of self-gravitating particles. The macroscopic Maxwell equations for a system of electromagnetically and gravitationally interacting particles also differ from the classical Maxwell equations in having additional terms in the left-hand side due to simultaneous effects described by general relativity and the interaction effects. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 107–131, October, 2000.     

A Survey of Quasi-Newton Equations and Quasi-Newton Methods for Optimization

Quasi-Newton equations play a central role in quasi-Newton methods for optimization and various quasi-Newton equations are available. This paper gives a survey on these quasi-Newton equations and studies properties of quasi-Newton methods with updates satisfying different quasi-Newton equations. These include single-step quasi-Newton equations that use only gradient information and that use both gradient and function value information in one step, and multi-step quasi-Newton equations that use the gradient information in last m steps. Main properties of quasi-Newton methods with updates satisfying different quasi-Newton equations are studied. These properties include the finite termination property, invariance, heredity of positive definite updates, consistency of search directions, global convergence and local superlinear convergence properties.     

Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

The article presents a new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations, including autonomous and nonautonomous ordinary differential equations (ODE), partial differential equations, and delay differential equations. The theory relies on four remarkable results: Feigenbaum’s period doubling theory for cycles of one-dimensional unimodal maps, Sharkovskii’s theory of birth of cycles of arbitrary period up to cycle of period three in one-dimensional unimodal maps, Magnitskii’s theory of rotor singular point in two-dimensional nonautonomous ODE systems, acting as a bridge between one-dimensional maps and differential equations, and Magnitskii’s theory of homoclinic bifurcation cascade that follows the Sharkovskii cascade. All the theoretical propositions are rigorously proved and illustrated with numerous analytical examples and numerical computations, which are presented for all classical chaotic nonlinear dissipative systems of differential equations.     

Invariant Simultaneous Solutions of Evolution Equations of Integrable Hierarchies

We construct classical point symmetry groups for joint pairs of evolution equations (systems of equations) of integrable hierarchies related to the auxiliary equation of the method of the inverse problem of second order. For the two cases: the hierarchy of Korteweg--de Vries (KdV) equations and of the systems of Kaup equations, we construct simultaneous solutions invariant with respect to the symmetry group. The problem of the construction of these solutions can be reduced, respectively, to the first and second Painlevé equations depending on a parameter. The Painlevé equations are supplemented by the linear evolution equations defining the deformation of the solution of the corresponding Painlevé equation.     

The Complete Solution of the Long-Wave–Short-Wave Resonance Equations

We give a complete analysis of the long-wave–short-wave resonance equations which appear in fluid mechanics as well as plasma physics. Using the inverse-scattering technique, these equations can be reduced to a pair of linear integral equations (Marchenko equations), with the N-soliton solutions intimately related to the asymptotic state of the evolution equations. The interaction of solitons and the conserved quantities are discussed.     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

On zeros of solutions of linear differential equations

We consider linear differential equations with regular coefficients in $\text{¦ z ¦ < 1}$. We obtain sufficient conditions for all the solutions of these equations to vanish a given number of times at the most. First the results are obtained for differential equations of second order, then for differential equations of nth order, n > 2.     

Mathematical and physical aspect of the solution of the Cauchy problem for a system of equations describing a nonlocal model of propagation of heat with finite speed

In our paper we present a new system of equations describing a nonlocal model of propagation of heat with finite speed in three-dimensional space. Such a system of equations is described by a system of integral – differential equations. At first using the modiffied Cagniard de Hoop method, we construct the fundamental solution of this system of equations. On the basis of the constructed fundamental solution we obtain the explicite formulate of the solution of the Cauchy problem for this system of equations and applying the method of Sobolev and Biesov spaces, we get L^p – L^q time decay estimate for the solution of the Cauchy problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Diagram equations of the theory of fully developed turbulence

Skeleton diagram equations of turbulence theory — the Dyson equations and the equations for vertices of three types — are obtained nonperturbatively. Their derivation is based on the use of an equation in functional derivatives for the characteristic functional of a hydrodynamic system described by Navier-Stokes equations in the presence of an external random force. The iterative solution of these equations reproduces the perturbation series for second moments that is usually obtained in a more complicated way and also the series for the third moments.Institute of Problems in Mechanics, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 1, pp. 28–37, October, 1994.     

Independence principles in the equations of state of a deformable material

We consider the principles of coordinate, rotational, and initial independence of the equations of state for a deformable material and the theorem on the existence of elasticity potential connected with them. We show that the well-known axiomatic substantiation and mathematical representation of these principles in “rational continuum mechanics” as well as the proof of the theorem are erroneous. A correct proof of the principles and theorem is presented for the most general case (a stressed anisotropic body under the action of an arbitrary tensor field) without applying any axioms. On this basis, we eliminated the dependence on an arbitrary initial state and the corresponding accumulated strain from the system of equations of state of a deformable material. The obtained forms of equations are convenient for constructing and analyzing the equations of local influence of initial stresses on physical fields of different nature. Finally, these equations represent governing equations for the problems of nondestructive testing of inhomogeneous three-dimensional stress fields and for theoretical-and-experimental investigation of the nonlinear equations of state.     

Convergence of a singular Euler-Poisson approximation of the incompressible Navier-Stokes equations

In this note, we rigorously justify a singular approximation of the incompressible Navier-Stokes equations. Our approximation combines two classical approximations of the incompressible Euler equations: a standard relaxation approximation, but with a diffusive scaling, and the Euler-Poisson equations in the quasineutral regime.

     

Homogenization of linear and nonlinear transport equations

We study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields. Two types of homogenized equations are derived. For general n-dimensional linear and nonlinear problems, we derive homogenized equations by introducing additional independent variables to represent the small scales. For the two-dimensional linear transport equations, we derive effective equations for the averaged quantities. Such equations take the form of either a degenerate non-local diffusion equation with memory or a higher order hyperbolic equation. To study the nonlinear transport equations we introduce the concept of two-scale Young measure and extend DiPerna's method to prove that it reduces to a family of Dirac measures.     

Asymptotic symmetry of solutions of nonlinear partial differential equations

For a large class of partial differential equations on exterior domains or on ?^N we show that any solution tending to a limit from one side as x goes to infinity satisfies the property of “asymptotic spherical symmetry”. The main examples are semilinear elliptic equations, quasilinear degenerate elliptic equations, and first-order Hamilton-Jacobi equations.