On Self-Similar Converging Shock Waves

Abstract

In this paper, we rigorously prove the existence of self-similar converging shock wave solutions for the non-isentropic Euler equations for \(\gamma \in (1,3]\). These solutions are analytic away from the shock interface before collapse, and the shock wave reaches the origin at the time of collapse. The region behind the shock undergoes a sonic degeneracy, which causes numerous difficulties for smoothness of the flow and the analytic construction of the solution. The proof is based on continuity arguments, nonlinear invariances, and barrier functions.