Three-dimensional Interaction of Shocks in Irrotational Flows

The general d-dimensional Riemann problem raises naturally the question of resolving the interaction of d planar shocks merging at a point. In gas dynamics, we may consider only standing shocks. This problem has received a satisfactory answer in dimension d = 2 (see [2, 3]). We investigate the 3-dimensional case. We restrict to the irrotational case, in order to keep the complexity of the solution within reasonable bounds. We show that a new kind of waves appears downstream, which we call a conical wave. When the equation of state is that of Chaplygin / von Kármán, we give a complete mathematical answer to this problem. This involves the existence and uniqueness of a complete minimal surface in a hyperbolic space, with prescribed asymptotics.