Diffraction of a shock into an expansion wavefront for the transonic self-similar nonlinear wave system in two space dimensions

Abstract

We consider a configuration where a planar shock reflects and diffracts as it hits a semi-infinite rigid screen. The diffracted reflected shock meets the diffracted expansion wave, created by the incident shock that does not hit the screen, and changes continuously from a shock into an expansion. The governing equation changes its type and becomes degenerate as the wave changes continuously from a shock to an expansion. Furthermore the governing equation has multiple free boundaries (transonic shocks) and an additional degenerate sonic boundary (the expansion wave). We develop an analysis to understand the solution structure near which the shock strength approaches zero and the shock turns continuously into an expansion wavefront, and show the existence of the global solution to this configuration for the nonlinear wave system. Moreover we provide an asymptotic analysis to estimate the position of the change of the wave, and present intriguing numerical results.