On the Hierarchy of Singularities in 3D Pressureless Gas

This paper discusses the process of shock waves propagation in three-dimensional isobaric media. The model for such media is a system of equations of gas dynamics, in which, formally, the pressure is assumed to be zero. Thus, the corresponding system of partial differential equations (PDEs) is often called the pressureless gas dynamics system. The general theory of systems of conservation laws states that the pressureless gas system is in some sense degenerate, and as a result, the corresponding generalized solutions may have strong singularities—developing shock waves with density in the form of delta functions on manifolds of different dimensions. This phenomenon is often referred to as the evolution of a hierarchy of strong singularities, or the evolution of a hierarchy of shock waves. Previous studies have mainly focused on singularity manifolds with codimension one, and some papers have explored the study of point singularities in two-dimensional case. However, non-trivial hierarchies of singularities should include a singularity manifold with a codimension lying between the minimum and maximum values. Three dimensions is the first dimension in which such an ’intermediate’ singularity can occur. In this paper, we study the laws of evolution for this type of singularity in three dimensions and obtain a corresponding system of PDEs. This system significantly differs from the systems for the evolution of maximal and minimal codimension singularities. We also demonstrate some examples of the behavior of ‘‘intermediate’’ singularities.