Transonic shock in finitely long nozzle with porous medium boundary condition

In this paper, we investigate the 2-D Euler equations with complex boundary conditions. For the entrance and exit of finitely long nozzle, we use the supersonic incoming flow condition and the end pressure condition, respectively. In addition, for the nozzle wall, the lower one is solid and the upper one is permitted to have leakage. We establish the well-posedness if appropriate seepage discharge is given. We also obtain the shock position by solving a free boundary problem without assuming the shock must go through a fixed point.