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A method for constructing geometric solutions of the Riemann problem for an impulsively perturbed conservation law is described. A complete classification of the possible patterns of the phase flow is given and, for each of the possible cases, the limit in the sense of Hausdorff is constructed.

All types of solutions to the Riemann problem for a system of conservation laws that is nonstrictly hyperbolic in the sense of Petrovskii, but not in the sense of Friedrichs are described.

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 existence of nonclassical (two-dimensional) shock waves in Riemann’s problem is proved for a modification of the system of shallow water equations.

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 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.

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.

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.

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

The paper is aimed at studying solvability conditions for the quadratic matrix Riccati equation that arises in connection with the Chapman–Enskog projection for the Cauchy problem and the mixed problem for moment approximations of kinetic equations. The structure of the matrix equation allows for the formulation of necessary and sufficient conditions for the existence of solutions in terms of eigenvectors and associated vectors of the coefficient matrix.

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.

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.

Solvability conditions are studied in this paper for a quadratic matrix Riccati equation arising in studies of the Chapman-Enskog projection for a Cauchy problem and a mixed problem for momentum approximations of kinetic equations. The structure of the matrix equation permits one to formulate necessary and sufficient solvability conditions in terms of eigenvectors and associated vectors for the matrix composed from the coefficients.

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 consider the Riemann’s problem for the multicomponent Euler system. In this paper, we describe possible types of shockwave and rarefaction wave bifurcations near the critical manifold.

The purpose of this paper is to investigate problems of the Navier-Stokes approximation to kinetic equations in terms of the so-called Chapman-Enskog projection. One considers properties of the Chapman-Enskog projection for the Cauchy problem for moment approximations of the kinetic equation and primarily the Chapman-Enskog projection for the Boltzmann-Peierls kinetic equation. The existence of the Chapman-Enskog projection for the Cauchy problem is proved for the phase space of conservative variables (phenomena of nonlinear diffusion) and for the phase space of physical variables (the second sound projection).

We construct a nonstandard regularization of the multispeed Euler system to reconstruct the base inner turbulence factors. We 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: 13 titles. Illustrations: 6 figures.