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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.

We construct solutions to the Cauchy problem for a model system that is not hyperbolic in the sense of Friedrichs. To this end, we apply a new geometric method for constructing solutions to the Riemann problem.

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.

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

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.

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

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.