Formation and propagation of singularities in one-dimensional Chaplygin gas

In this paper, we investigate the formation and propagation of singularities for the system for one-dimensional Chaplygin gas, which is described by a quasilinear hyperbolic system with linearly degenerate characteristic fields. The phenomena of concentration and the formation of ``$\delta$-shock'' waves are identified and analyzed systematically for this system under suitably large initial data. In contrast to the Rankine-Hogoniot conditions for classical shock, the generalized Rankine-Hogoniot conditions for ``$\delta$-shock'' waves are established. Finally, it is shown that the total mass and momentum related to the solution are independent of time.