Formation of singularities in solutions to ideal hydrodynamics of freely cooling inelastic gases

TL;DR

This paper proves that solutions to ideal granular hydrodynamics equations typically develop singularities in finite time, especially for certain adiabatic indices, and constructs explicit solutions demonstrating initial singularities.

Contribution

It establishes finite-time singularity formation in ideal granular hydrodynamics and constructs explicit solutions with initial singularities, advancing understanding of solution behavior.

Findings

Solutions lose smoothness in finite time for certain parameters.

Explicit solutions with initial singularities are constructed.

Singularity formation is proven in multiple spatial dimensions.

Abstract

We consider solutions to the hyperbolic system of equations of ideal granular hydrodynamics with conserved mass, total energy and finite momentum of inertia and prove that these solutions generically lose the initial smoothness within a finite time in any space dimension $n$ for the adiabatic index $\gamma \le 1+\frac{2}{n}.$ Further, in the one-dimensional case we introduce a solution depending only on the spatial coordinate outside of a ball containing the origin and prove that this solution under rather general assumptions on initial data cannot be global in time too. Then we construct an exact axially symmetric solution with separable time and space variables having a strong singularity in the density component beginning from the initial moment of time, whereas other components of solution are initially continuous.