On the two-dimensional gas expansion for compressible Euler equations

We investigate the problem of two-dimensional, unsteady expansion of an inviscid, polytropic gas, which can be interpreted as the collapse of a wedge-shaped dam containing water initially with a uniform velocity. We model this problem by isentropic Euler equations. The flow is quasi-stationary, and using hodograph transform, we describe it by a partial differential equation of second order in the state space if it is irrotational initially. Furthermore, this equation is reduced to a linearly degenerate system of three partial differential equations with inhomogeneous source terms. These properties are used to pro e that the flow is globally smooth when a wedge of gas expands into a vacuum, and to analyze that shocks may appear in the interaction of four planar rarefaction waves.