Discontinuous solutions to the Euler equations for a van der Waals fluid

In this work we study the discontinuous solutions to the Euler equations for a van der Waals fluid, which contain one shock and one phase transition. We consider the general case when there is a characteristic between the shock front and the phase boundary. We establish the local existence of such solutions.     

Selfsimilar solutions of the problem of heat and mass transfer in a saturated porous medium with a volume heat source

Selfsimilar solutions of a system of stationary equations of heat condunction and filtration of molten material in the presence of a volume heat source generated by absorption of the energy of electromagnetic radiation, are considered. The possibility of the existence of a self-similar solution in the case of various (plane, cylindrical and spherical) spatial symmetries is studied. The existence of a selfsimilar solution is shown for the axisymmetric case when the radiation obeys a prescribed law. The influence of the surface volume heating and convective heat transfer due to filtration is studied. A solution for the case when the filtration of the molten phase is quasistationary is also investigated.     

Equations for the continuous estimation of the perturbations of dynamical systems

A dynamical system acted upon by external perturbations is considered. It is assumed that the phase state of the system (or a part of it) is observed with certain errors. The problem is to construct differential equations for estimating (reconstructing) the perturbations using measurement data. Unlike in papers in which cases of discrete instants of the observations are analysed, the continuous case is considered for which differential equations of an auxiliary system are derived, the controls in which are approximations of an unknown input. The general constructions are illustrated by means of an example.     

Weak and Strong Interactions between Internal Solitary Waves

The interaction of weakly nonlinear long internal gravity waves is studied. Weak interactions occur when the wave phase speeds are unequal; this case includes that of a head-on collision. It is shown that each wave satisfies a Korteweg-de Vries equation, and the main effect of the interaction is described by a phase shift. Strong interactions occur when the wave phase speeds are nearly equal although the waves belong to different modes. This case is described by a pair of coupled Korteweg-de Vries equations, for which some preliminary numerical results are presented.     

Poisson variations in the problem of the stability of equilibria in rigid body mechanics

The existence and stability of equilibria in rigid body mechanics is considered. A class of variations is indicated which satisfy the analogue of Poisson's equations, suitable for use when investigating both the sufficient and necessary conditions for the stability of such equilibria and which, in particular, enable the invariance of the equations of motion of the system and their first integrals to be effectively used when the phase variables and parameters of the problem are interchanged. The result is illustrated using the example of the problem of the motion of a gyrostat far from attracting objects.     

On a control problem for a nonlinear second-order system

A control problem for a nonlinear second-order system of differential equations in the presence of uncontrollable effects is investigated. A solution algorithm is proposed in the case when one phase coordinate of the system is measured at discrete moments. The algorithm is stable with respect to information noises and computational errors. Results of a computer experiment are presented.     

A representation of the equations of multicomponent multiphase seepage

A mathematical model of non-isothermal multicomponent flows in a porous medium is investigated. A general case is considered when the model can be used to describe processes with an arbitrary number of components and phases. A general form of the system of mixed-type equations describing the flow, which is similar to the Godunov form for hyperbolic systems is proposed. The equations obtained are applicable to flows with gas, liquid and solid phases. The thermodynamic properties of the medium are determined solely by a single multivalued function, by changing which one can obtain models of different flows in a porous medium. A clear geometrical interpretation of the solutions of the equations is proposed. An equation for the entropy is obtained, and it is shown that in order that the model should not contradict the second law of thermodynamics, it is necessary to take into account, in the energy equation, the work of the gravity force, which is often neglected when investigating seepage.     

An algebraic theory of strong power in negatively connected exchange networks

Various extensions of the model are proposed to deal with a wider variety of conditions than are normally examined in experiments on exchange networks. With little or no modification, the model can predict power when exchange relations are unequal in value, when positions vary in the number of exchanges in which they can participate, and when three or more participants are required for a transaction to occur.

A structural and algebraic theory of power in negatively connected exchange networks can be deduced from a few simple and plausible assumptions about how individuals make decisions. The model generates a set of equations. A typology of exchange networks follows from characteristics of the solution to these equations. There are four possibilities: the equations have a unique solution in which some positions have all the power; the equations have a unique solution in which all positions have equal power; the equations have an infinity of solutions, in which case power is undetermined by structural considerations; the equations have no solution, in which case power should be unstable.     

Diffusion processes in linear spaces and pseudotopologies

The theory of diffusion processes with a nonnormable phase space (a nuclear Fréchet space) is developed and the Cauchy problem for parabolic equations relative to functions on this space is solved by probabilistic methods. A series of examples are given, demonstrating a significant difference between the theory of stochastic differential equations and parabolic equations in the case of locally convex spaces, on one hand, and the analogous theory in the case of Banach spaces, on the other hand. The difficulty which arises, when passing from a Banach space to a Fréchet space, involves basically a functional rather than a probabilistic character. There appears a sufficiently complex intertwinement of the theory of locally convex and pseudotopological spaces with probability theory.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 190–209, 1986.In conclusion, the author expresses his gratitude to O. G. Smolyanov for his constant interest in the paper and for useful advice.     

Homoclinic Solutions and Chaos in Ordinary Differential Equations with Singular Perturbations

Ordinary differential equations are considered which contain a singular perturbation. It is assumed that when the perturbation parameter is zero, the equation has a hyperbolic equilibrium and homoclinic solution. No restriction is placed on the dimension of the phase space or on the dimension of intersection of the stable and unstable manifolds. A bifurcation function is established which determines nonzero values of the perturbation parameter for which the homoclinic solution persists. It is further shown that when the vector field is periodic and a transversality condition is satisfied, the homoclinic solution to the perturbed equation produces a transverse homoclinic orbit in the period map. The techniques used are those of exponential dichotomies, Lyapunov-Schmidt reduction and scales of Banach spaces. A much simplified version of this latter theory is developed suitable for the present case. This work generalizes some recent results of Battelli and Palmer.

     

On a comparison method for pulse systems in the space Rn

A method for the study of differential equations with pulse influence on the surfaces, which was realized in [1] for a bounded domain in the phase space, is now extended to the entire spaceR ⁿ. We prove theorems on the existence of integral surfaces in the critical case and justify the reduction principle for these equations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 753–762, June, 1993.     

Splitting superfluid hydrodynamic equations and instability of solutions in the case of degenerate bosons

Properties of solutions to superfluid hydrodynamic equations as applied to the degenerate Bose gas are considered. The equations are split into two independent pairs of equations. One pair is written for the normal component implies the instability of solutions, which manifests itself in the majorant catastrophe with respect to the total density. The case when the thermodynamic functions depend on the difference of the normal and superfluid velocities is also considered. In that case, the system is not split; however, the instability and the majorant catastrophe occur when the initial temperature tends to absolute zero.     

Instability of solitary waves in non-linear composite media

A problem on the dynamic instability of soliton solutions (solitary waves) of Hamilton's equations, describing plane waves in non-linear elastic composite media with or without anisotropy, is considered. In the anisotropic case, there are two two-parameter families of solitary waves: fast and slow and, when there is no anisotropy, there is one three-parameter family. A classification of the instability of solitary waves of the fast family in the anisotropic case and of representatives of families of solitary waves, the velocities of which lie outside of the range of stability when there is anisotropy and when there is no anisotropy, is given on the basis of a numerical solution of a Cauchy problem for the model equations. In this paper, the fundamental equations describing plane waves in non-linear, anisotropic, elastic composites are derived, the Hamilton form of the basic equations is presented, the symmetries in the anisotropic and isotropic cases are considered, the conserved quantities and the soliton solutions, that is, the solitary waves are presented, the nature of the instability of representatives of all three families is investigated, brief formulation of the results is presented and problems of the instability of the fast family in the anisotropic case and of representatives of the families, the velocities of which lie outside of the range of stability in the presence and absence of anisotropy (explosive instability), are discussed.     

The propagation of weak concentration discontinuities in the isothermal flow of a multicomponent mixture in a porous medium with phase transitions

It is shown that for the general case of a system of non-linear equations, describing multicomponent isothermal flow in a porous medium with phase transitions, as in hyperbolic systems, weak concentration discontinuities propagate with finite velocities, which are determined by solving an eigenvalue problem. If the seeping phases are incompressible and there are no phase transitions, the results obtained for weak discontinuities transfer into the well-known formulae for the Buckley – Leverett model. The results are demonstrated for the case of two-component seepage with phase transitions.     

The gap phenomenon in the dimension study of finite type systems

Several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the symmetries of geometric structures and differential equations. A general result clarifying this effect in the case when the structure is associated to a vector distribution, is proposed.     

On a control algorithm for a linear system with measurements of a part of coordinates of the phase vector

We consider a feedback control problem for a system of ordinary differential equations in the case when only a part of coordinates of the phase vector are measured and propose a solution algorithm that is stable to perturbations. The algorithm is based on a combination of the theories of dynamical inversion and guaranteed control. It consists of two blocks: a block for the dynamical reconstruction of unmeasured coordinates and a control block.     

Homogenization and convergence of the vortex method for 2-D euler equations with oscillatory vorticity fields

The 2-D incompressible Euler equations with oscillatory vorticity fields are studied. A homogenization result for 2-D Euler equations in velocity-vorticity formulation is obtained and weak continuity of the equations is proved. Convergence of the vortex method is analyzed in the case when the continuous vorticity is not well resolved by the computational particles. Numerical results are given. Comparisons are made with the corresponding finite difference approximation.     

Convergence of the semi-implicit Euler method for neutral stochastic delay differential equations with phase semi-Markovian switching

Recently, in the numerical analysis for stochastic differential equations (SDEs), it is a new topic to study the numerical schemes of neutral stochastic functional differential equations (NSFDEs) (see Wu and Mao [1]). Especially when Markovian switchings are taken into consideration, these problems will be more complicated. Although Zhou and Wu [2] develop a numerical scheme to neutral stochastic delay differential equations with Markovian switching (short for NSDDEwMSs), their method belongs to explicit Euler–Maruyama methods which are in general much less accurate in approximation than their implicit or semi-implicit counterparts. Therefore, to propose an implicit method becomes imperative to fill the gap. In this paper we will extend Zhou and Wu [2] to the case of the semi-implicit Euler–Maruyama methods and equations with phase semi-Markovian switching rather than Markovian switching. The employment of phase semi-Markovian chains can avoid the restriction of the negative exponential distribution of the sojourn time at a state. We prove the semi-implicit Euler solution will converge to the exact solution to NSDDEwMS under local Lipschitz condition. More precise inequalities and new techniques are put forward to overcome the difficulties for the existence of the neutral part.     

On the Number of Conserved Quantities for the Two-Layer Shallow-Water Equations

The shallow-water equations for two-layer inviscid flow with a free surface overlying a rigid horizontal bottom subject to gravitational forcing only are examined to determine the possible forms of conservation laws that the equations permit. In the case of a single layer with flow in only one horizontal direction, it is known that there are an infinite number of associated equations in conservation form, where the conserved quantity is a multinomial in the layer variables. The method used to determine this result is generalized to show that in the two-layer case, the result does not generalize, and it is discovered that only a finite number of conservation equations exist when the density difference between the layers is nonzero. The subsequent conservation equations are given explicitly, and a systematic method for deriving conservation laws from an arbitrary first-order system is described. For the case when the flow is in both horizontal dimensions, the method of analysis is straightforward in the one-layer case, and the finite number of conservation equations are derived. The two-layer case is similar, and the finite number of generalized conserved quantities are stated, although the question of whether or not there are only a finite number is posed as an open question.