On the evolution of the hierarchy of shock waves in a two-dimensional isobaric medium

Abstract

In the proposed paper, the process of propagation of shock waves in two-dimensional media without its own pressure drop is studied. The model of such media is a system of equations of gas dynamics, where formally the pressure is assumed to be zero. From the point of view of the theory of systems of conservation laws, the system of equations under consideration is in some sense degenerate, and, consequently, the corresponding generalized solutions have strong singularities (evolving shock waves with density in the form of delta functions on manifolds of different dimensions). We will denote this property as the evolution of the hierarchy of strong singularities or the evolution of the hierarchy of shock waves. In the paper, in the two-dimensional case, the existence of such an interaction of strong singularities with density delta function along curves in the space $\mathbb{R}^2$ is proved, at which a density concentration occurs at a point, that is, a hierarchy of shock waves arises. The properties of such dynamics of strong singularities are described. The results obtained provide a starting point for moving on to a much more interesting multidimensional case in the future.