  We employ granular hydrodynamics to investigate a paradigmatic problem of
clustering of particles in a freely cooling dilute granular gas. We consider
large-scale hydrodynamic motions where the viscosity and heat conduction can be
neglected, and one arrives at the equations of ideal gas dynamics with an
additional term describing bulk energy losses due to inelastic collisions. We
employ Lagrangian coordinates and derive a broad family of exact non-stationary
analytical solutions that depend only on one spatial coordinate. These
solutions exhibit a new type of singularity, where the gas density blows up in
a finite time when starting from smooth initial conditions. The density blowups
signal formation of close-packed clusters of particles. As the density blow-up
time $t_c$ is approached, the maximum density exhibits a power law $\sim
(t_c-t)^{-2}$. The velocity gradient blows up as $\sim - (t_c-t)^{-1}$ while
the velocity itself remains continuous and develops a cusp (rather than a shock
discontinuity) at the singularity. The gas temperature vanishes at the
singularity, and the singularity follows the isobaric scenario: the gas
pressure remains finite and approximately uniform in space and constant in time
close to the singularity. An additional exact solution shows that the density
blowup, of the same type, may coexist with an "ordinary" shock, at which the
hydrodynamic fields are discontinuous but finite. We confirm stability of the
exact solutions with respect to small one-dimensional perturbations by solving
the ideal hydrodynamic equations numerically. Furthermore, numerical solutions
show that the local features of the density blowup hold universally,
independently of details of the initial and boundary conditions.
