On the expansion of a wedge of van der Waals gas into a vacuum

Abstract

We study the existence of global in time classical solution to the expansion of a wedge of van der Waals gas into a vacuum. We reduce this problem to a Goursat-type boundary value problem for 2D self-similar Euler system. The 2D self-similar Euler system is a mixed type system, the type in each point is determined by the local pseudo-Mach number. By introducing the concept of invariant square region of solution, we prove that this system is strictly hyperbolic in the flow region of the Goursat problem. A prior C1 estimate of the solution to the Goursat problem is obtained by using the method of characteristic decomposition. Due to the existence of vacuum boundary, the classical approach to extend local solution to global solution does not work here. We extend the local solution of the Goursat problem up to the interface of gas with vacuum by solving many “small” Goursat problems in each extension step.