On Model Two-Dimensional Pressureless Gas Flows: Variational Description and Numerical Algorithm Based on Adhesion Dynamics

Abstract

Weak solutions of the system of pressureless gas dynamics equations in two dimensions are studied. Theoretical issues are considered, namely, the general mathematical theory of conservation laws for the system is addressed. Emphasis is placed on an important distinctive feature of this system: the emergence of strong density singularities along manifolds of different dimensions. This property is characterized as the formation of a hierarchy of singularities. In earlier application-oriented works (e.g., by A.N. Krayko, et al., including more complicated cases of 3D two-phase flows), this property was studied at the physical level of rigor. In this paper, the formation of a hierarchy of singularities is examined mathematically, since, for example, the existence of a solution with a strong singularity at a point (in the 2D case) is rather difficult to prove rigorously. Accordingly, a special numerical algorithm is used to develop mathematical hypotheses concerning solution behavior. Approaches to the construction of a variational principle for weak solutions are considered theoretically. A numerical algorithm based on approximate adhesion dynamics in the multidimensional case is implemented. The algorithm is tested on several examples (2D Riemann problem) in terms of internal convergence and is compared with mathematical results, including those obtained by other authors.