Time-periodic transonic shock solution in divergent nozzles

We demonstrate that it is possible to control a normal transonic shock to move periodically by adjusting the boundary conditions at the entrance or the exit of the tube, for which, the phenomena has been observed in engineering. In this paper, we describe the gas by a quasi-one-dimensional compressible Euler equations with temporal periodic boundary conditions and prove the global existence and dynamical stability of the time-periodic transonic shock solution with an iteration method. The major difficulty is to determine the position of the moving shock front, which can be obtained by a free boundary problem in the subsonic domain. We decouple this free boundary problem by the $Rankine-Hugoniot$ conditions and a two-step iteration process.