Exact solutions with singularities to ideal hydrodynamics of inelastic gases

TL;DR

This paper derives a broad class of exact solutions for ideal granular hydrodynamics, revealing conditions for smoothness or singularity formation, including special cases like the Chaplygin gas, with complex behaviors in 1D and 2D.

Contribution

It constructs explicit solutions to multidimensional ideal granular hydrodynamics equations for arbitrary adiabatic index, including singularity formation scenarios and special reductions for specific gamma values.

Findings

Solutions can remain smooth or develop singularities depending on initial conditions.

Singularities can form at points or lines in 2D cases.

Special case for gamma = -1 reduces to a system related to Chaplygin gas.

Abstract

We construct a large family of exact solutions to the hyperbolic system of 3 equations of ideal granular hydrodynamics in several dimensions for arbitrary adiabatic index $\gamma$. In dependence of initial conditions these solutions can keep smoothness for all times or develop singularity. In particular, in the 2D case the singularity can be formed either in a point or along a line. For $\gamma=-1$ the problem is reduced to the system of two equations, related to a special case of the Chaplygin gas. In the 1D case this system can be written in the Riemann invariant and can be treated in a standard way. The solution to the Riemann problem in this case demonstrate an unusual and complicated behavior.