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The author’s method of thermodynamic analysis is used to single out two equations of state for the laminar combustion process: the classical Hugoniot adiabat, which determines the pressure, and the equation of state, which determines the entropy. This allows constructing a new mathematical model of the laminar process of vibrational combustion of a two-component mixture by closing the classical models of continuum mechanics. The model is phenomenological, which requires its verification. For numerical verification, the well-known experimental fact is chosen, the appearance of high-frequency acoustic vibrations described by B.V. Raushenbakh. The conditions for the origin of high-frequency oscillations are obtained in terms of the standard chemical potential. They can substantially disturb the combustion process and may cause a catastrophic break-up of the furnace of the engine structure. A numerical experiment established critical values of the standard chemical potential when high-frequency vibrations lead to destruction.

A new model of laminar vibrational combustion is constructed relying on a thermodynamic analysis of the combustion process. Two combustion modes, detonation and deflagration, are modeled. The nature of their origin depending on the structure of the standard chemical potential is established, and a numerical experiment concerning the onset of these combustion modes is carried out. By controlling the velocity of the passive component at the inlet, the critical value of the standard chemical potential is identified at which there appear high-frequency vibrations leading to blow-up.

This article is an attempt to study the process of Rayleigh–Benard convective instability by the methods used for mathematical modeling of critical phenomena as nonequilibrium phase transitions in their initial stages of spinodal decomposition. We show that it is possible to extend the formalism adopted in the Cahn–Hillard theory of nonequilibrium phase transitions and perfected on problems of highgradient crystallization to other types of problems, in particular, those pertaining to the Rayleigh–Benard convective instability. For the initial stage of instability, a model is constructed that represents it as a nonequilibrium phase transition due to diffusive stratification. It is shown that the Gibbs free energy of deviation from the homogeneous state (with respect to the instability under consideration) is an analogue of the Ginsburg–Landau potential. Numerical experiments, by means of boundary temperature control, have been conducted with regard to self-excitation of the homogeneous state. Numerical analysis shows that convective flows may appear and proceed from regular forms (the so-called regular structures) to nonregular flows through a chaotization of the process. External factors, such as temperature growth, may lead to chaos via period doubling bifurcations.

A model was constructed for reconstructing the initial stage of solidification of binary alloys treated as a nonequilibrium phase transition with a diffusion stratification mechanism. Numerical experiments concerning the self-excitation of a homogeneous state by applying a melt cooling boundary control condition were performed.

For laminar-turbulent transition model is built reconstruction of the initial stage of instability as a nonequilibrium phase transition, the mechanism of which is diffusion stratification. It is shown that the Gibbs free energy deviations from the homogeneous state (relative to the instability under consideration) is an analogue Ginzburg-Landau potentials. Numerical experiments were performed. Self-excitation of a homogeneous state by edge control condition of increasing speed. Under external influence (increase in speed at the input), there is a transition to chaos through bifurcations of period doubling, when the internal control parameter (analogue of the Reynolds number) changes, like the Feigenbaum period doubling cascade.

We consider one-dimensional Carleman and Godunov–Sultangazin equations and obtain local equilibrium conditions for solutions of the Cauchy problem with finite energy and periodic initial data. Moreover, we prove the exponential stabilization to the equilibrium state.

Sufficient conditions are found for the existence of stabilizing solutions of the Riccati differential equation y′ = (y − y1(x)) (y − y2(x)) with given y1(x) and y2(x). For various types of stabilizing solutions, the number of points of extremum is examined.

The initial stage of instability in the failure of a structural material treated as a nonequilibrium phase transition is reconstructed. Its mechanism is based on spinodal decomposition (diffusion separation).

The initial stage of the Rayleigh–Benard convective instability regarded as a nonequilibrium phase transition is reconstructed. Its mechanism is based on spinodal decomposition (diffusion separation).

We study local equilibrium of solutions to the Cauchy problem with bounded energy and generation of chaotic dynamics.

The initial stage of the laminar–turbulent transition is reconstructed. Its mechanism is based on spinodal decomposition (diffusion separation).

We construct a nonstandard regularization for a multicomponent Euler system and obtain analogs of the Hugoniot condition and the Lax stability condition. We investigate the local accessibility problem for phase space points and construct dual bifurcations of one-front solutions of the truncated Euler system into two-front solutions.

For the laminar–turbulent transition, we construct a model of reconstruction of the initial stage of instability qua a nonequilibrium transition with diffusion separation mechanism. It is shown that the free Gibbs energy of departure from the homogeneous state (with respect to the instability under consideration) is an analogue of the Ginzburg–Landau potential. Numerical experiments for self-excitation of the homogeneous state with control of the boundary condition of velocity increase were carried out, which showed the appearance of the laminar–turbulent transition and its development from regular forms (the so-called dissipative structures) with subsequent transition to irregular flows via chaotization of the process. An external action (an increase in velocity) results in a transition to chaos in terms of period-doubling bifurcations similarly to the Feigenbaum cascade of period-doubling bifurcations. The chaotization of the process transforms regular forms (dissipative structures) into the two-velocity regime (the regime of two shock waves), which was called the Riemann–Hugoniot catastrophe by Prigogine and Nicolis. This transformation depends substantially on gravitation. The perturbation is shown to be nonlocal, which indications that the classical perturbation theory is inapplicable in this case.

Based on the Bloch principle, we obtain local equilibrium conditions for solutions to the Cauchy problem with bounded energy for the one-dimensional Carleman equation in the weighted L2 spaces. Bibliography: 5 titles. Illustrations: 1 figure.

For the truncated Euler system, we study the problem of local reachability of points of the state space. We construct bifurcations of one-front solutions of the truncated Euler system into two-front solutions. The truncated Euler system is an example of a nonstrictly hyperbolic system of conservation laws for which there is no complete basis of eigenvectors on the critical manifold (of multiple eigenvalues) and there exists an associated vector. The constructed bifurcations of critical shock waves give an answer to the Lax problem on the behavior of a shock wave after it passes through the critical manifold in the phase space.

UDC 517.9 We consider the Cauchy problem for the one-dimensional Carleman equation with bounded energy and periodic initial data and obtain the local equilibrium conditions. We prove exponential stabilization to the equilibrium state. Bibliography :1 0titles. Illustrations :3 figures. Dedicated to N. N. Uraltseva

The hypothesis of two-speed hydrodynamics at the initial stage of formation of turbulence is investigated.

The initial phase of instability in the structure of a solid in the form of a fracture process of the structure material as a nonequilibrium phase transition which depends on the spinodal decomposition (diffusion separation into two pases with different degradation factor) is reconstructed.The initial phase of instability in the structure of a solid in the form of a fracture process of the structure material as a nonequilibrium phase transition which depends on the spinodal decomposition (diffusion separation into two pases with different degradation factor) is reconstructed.

In the paper we study the reproducing of the initial phase of the inner turbulence (without regard for the boundary effects). The atypical regularization of multiple-component Euler system is made by the viscosity and diffuse layering introduction. The analogue of Hugoniot condition and the analogue of Lax stability condition are constructed for it. The problem of local accessibility of the phase space points is investigated. The bifurcations of one-front solutions of the abridged Euler system to the two-front solutions are obtained. The supersonic behaviour of bifurcations appearance is shown. The reconstruction of the initial phase of the inner turbulence (without regard for the boundary effects) is made including the mathematical description of the birth of two-speed flow (the Riemann-Hugoniot catastrophe) and alternation.

Existence of global solution for a 2-dimensional discrete equation of kinetics and expansion with respect to smoothness are obtained, and the effect of progressing waves generated by the operator of interaction is investigated.

We consider the truncated Euler system and study the existence of bifurcations of critical rarefaction waves. Bibliography: 6 titles.

We construct a nonstandard regularization of the multicomponent Euler system and obtain counterparts of the Hugoniot condition and the Lax stability condition.

We consider the truncated Euler system and analyze bifurcations of a one-front solution to a solution with two fronts. In particular, it is shown that bifurcations of double-humped (single-humped) kink type are caused by perturbations of the upper (lower) critical solution. Bibliography: 2 titles.

It is proved, for a four-moment model of a phonon gas system and the Dirac–Schwindler extension of the Maxwell system, that a Chapman correct restriction of a initial-boundary problem exists. The well-posedness condition is found in terms of algebraic relations for parameters of the problem and elements of the boundary matrix.

The existence of a global solution to the discrete kinetic equations in Sobolev spaces is proved, its decomposition by summability is obtained, the influence of its oscillations generated by the interaction operator is explored. The existence of a submanifold Mdiss of initial data (u0,v0,w0) for which the dissipative solution exists is proved. It’s shown that the interaction operator generates the solitons (progressive waves) as the nondissipative part of the solution when the initial data (u0,v0,w0) deviate from the submanifold Mdiss. The amplitude of solitons is proportional to the distance from (u0,v0,w0) to the submanifold Mdiss. It follows that the solution can stabilize as t→∞ only on compact sets of spatial variables.

Our study concerns the existence of a global solution to the discrete kinetic equations in Sobolev spaces. We obtain a decomposition of the solution and clarify the influence of the oscillations generated by the interaction operator. We show that there exists a submanifold of initial data for which dissipative solution exit. If initial data u 0, v 0, w 0 deviate from the submanifold , then the interaction operator generates solitons, the nondissipative part of the solution. The amplitude of solitons is proportional to the distance from u 0, v 0, w 0 to the submanifold , which means that the solution can stabilize as t→∞ only on compact sets of spatial variables. Bibliography: 6 titles.

The paper reconstructs the early stage of the Rayleigh–Benard convective instability considered as a nonequilibrium phase transition with the spinodal decomposition (diffusion separation) mechanism.

The complexity of problems of the theoreticaldescription of the crystallization process was noted bymany experts. For example, E.N. Kablov stated thefollowing in [1, p. 300] when considering the directedcrystallization process of heatresistant nickel alloys:“Description of the morphology of the phase boundary when taking into account the instabilities that ariseis very difficult even for singlephase alloys.”N.P. Lyakishev and G.S. Burkhanov [2, p. 140] adhereto the same opinion: “Until now there has been nosufficiently complete theory of crystallization ofeutectic alloys explaining the formation of variouseutectic structures.” According to [2], the first attemptat explaining the formation mechanism of lamellareutectics was undertaken by Tamman, while Fogelassumed that both phases grow simultaneously, and, inthis case, their common interfaces should be perpendicular to the interface between solid and liquidphases. Vast experimental material on the problem offormation of various structures upon crystallizationfrom solutions and melts including the description ofthe wellknown Lisegang structures and attempts atconstructing a theory of similar phenomena considered in monograph [3] of F.M. Shemyakin andP.F. Mikhalev had been accumulated already by 1938.According to [3], it was G.V. Woolf who in 1926 proposed for the first time to consider the formation of analloy crystal structure of eutectic composition as anunstable process accompanied by the alternate formation of crystallites of one component or another.Thus, this problem has been formulated over a longtime, and all available means have been used for itssolution. It is of interest to note that, for describing thealternation of rings formed during the crystallizationof a deposit from solution (Lisegang structures), Susann Weil (1931) proposed to use the formulas following from quantum mechanics, where the ring numberis an analogue of the quantum number. In this case,good agreement between the calculated and experimental results [3] was obtained.Nowadays, it is possible to consider as an established fact that the complex structures arising duringcrystallization, for example, alloys, are a consequenceof the development of certain instabilities during thephase transition. However, there are many causes forthe occurrence of instabilities and, as experimentalinvestigations show, various variants of their development can be implemented. Without claiming completeness, it is possible to list the following possibilitiesleaving the priority for the development of the phasetransition instability:(i) concentration overcooling;(ii) convective flows deforming the temperaturefield (gravitational and thermocapillary convection);(iii) phase stratification.In addition, the elastic properties of the solid phase[2] (one example is epitaxial crystallization on substrates [4, 5]) as well as the thin structure of the phaseboundary (atomically smooth or diffuse surfaces [1,p. 298]) and the adsorption phenomena [6] also give acertain contribution. The adsorption phenomenadetermine the surface energy (surface tension), wetting, and spreading and, thus, can cause thermocapillary convection [7–9]. Nevertheless, nowadays there isno general approach to description of the causes of theoccurrence of all variants of structures arising duringthe solidphase formation from melt or solution,which are a consequence of the instability of the processes accompanying the phase transition. Materialscience achieved a significant success, especially anapplied one, but the further development of this fieldof science determining the technical progress in many

We establish the existence of global solutions to discrete kinetic equations in Sobolev spaces. We also obtain an expansion of the solution and investigate the influence of oscillations generated by the interaction operator. Bibliography: 7 titles.

A mathematical model is proposed to describe the process of crystallization. This model reflects experimentally established properties of structure formation in instability zones. Contemporary approaches to the theoretical description of crystallization yield values of the solid phase growth rate that differ, by several orders, from those observed in experiments. This fact stimulated the construction of a mathematical object that reproduces the basic instabilities of the process and stabilizes their feedback. The creation of this object required matching between the micro and the macro scales, as well as between wave and diffusion processes. The paper also gives a physical interpretation and results of the numerical analysis of the said model.

Two mathematical crystallization models describing structure formations in instability zones are proposed and justified. The first model, based on a phase field system, describes crystallization processes in binary alloys. The second model, based on a modified Biot model of a porous medium and the convective Cahn-Hilliard model, governs oriented crystallization. Physical interpretation and numerical analysis are discussed. Bibliography: 23 titles. Illustration: 5 figures.

The problems of the kinetics for hyperbolic regularizations of conservation laws are studied.

We study the large-time behaviour of global smooth solutions to the Cauchy problem for hyperbolic regularization of conservation laws. An attracting manifold of special smooth global solutions is determined by the Chapman projection onto the phase space of consolidated variables. For small initial data we construct the Chapman projection and describe its properties in the case of the Cauchy problem for moment approximations of kinetic equations. The existence conditions for the Chapman projection are expressed in terms of the solvability of the Riccati matrix equations with parameter.

This monograph consists of two volumes and is devoted to second-order partial differential equations (mainly, equations with nonnegative characteristic form). A number of problems of qualitative theory (for example, local smoothness and hypoellipticity) are presented.

We study the large-time behavior of global smooth solutions to the Cauchy problem for hyperbolic regularization of conservation laws. An attracting manifold of special smooth global solutions is determined by the Chapman-Enskog projection onto the phase space of consolidated variables. For small initial data we construct the Chapman-Enskog projection and describe its properties in the case of the Cauchy problem for moment approximations of kinetic equations. The existence conditions for the Chapman-Enskog projection are expressed in terms of the solvability of the Riccati matrix equations with parameter. Bibliography: 21 titles.

One of the problems of the kinetics of nonequilibrium processes is related to the lack of information concerning most of the nonequilibrium variables, namely, those which have no intuitive physical meaning, i.e., cannot be defined from the experiment. Moreover, the number of nonequilibrium variables is so large that a reasonable amount (from the physical point of view) of boundary conditions is insufficient for posing the mixed problem. What do the initial data for the Cauchy problem and the boundary conditions for the mixed problem mean in this case? In fact, we must assume that the initial-boundary data for most of the nonequilibrium variables (the higher-order momenta) are arbitrary! The British physicists Chapman and Enskog conjectured that, for “physically correct” models of continuum mechanics, the influence of the higher-order momenta is “inessential.” There are some postulates of physical correctness, but we do not dwell on them. For us it is of importance to understand what the fact that the influence of the higher-order momenta is “inessential” means from the mathematical point of view. The paper is devoted to this very topic.

We formulate conditions for the existence of a Chapman-Enskog projection in terms of the solvability of certain matrix equations, and obtain necessary and sufficient conditions for the existence of a solution to these equations.

The initial stage of laminar-turbulent transition reconstruction, the mechanism of which is the spinodal fission (or diffusive bundle), is considered.

International audience; For some qualitatively new problems with free (unknown) boundary, conditions of local solvability (with respect to time) are investigated.