Emergence of a Hierarchy of Singularities in Zero-Pressure Media. Two-Dimensional Case

Generalized solutions of the system of zero-pressure gas dynamics equations in the case of two spatial variables are considered. In contrast to the much-studied case of one spatial variable, the two-dimensional situation, as well as the multidimensional situation in general, is characterized by the fact that strong singularities can arise on manifolds of various dimensions. This property will be referred to as the existence of a hierarchy of strong singularities. We show that the generalization of the Rankine–Hugoniot relations must be extended in the presence of a hierarchy of singularities and give the form of such an extension. We use the Riemann initial data as an example to show how to construct a generalized solution in the case of a hierarchy of singularities.