Subsonic time-periodic solution to damped compressible Euler equations with non-dissipative boundary on one side

In this paper, we are interested in the time-periodic solutions of one-dimensional compressible Euler equations with linear damping α(x)ρ u. The boundary conditions have no dissipative structure on one side, which involves some physical reality. By using a delicately designed iteration scheme, we establish the existence and stability of time-periodic solutions on the perturbation of a subsonic Fanno flow. Moreover, if the boundary has a higher regularity, we show that the time-periodic solution has the same higher regularity and stability. Our results are also valid for Euler equations without damping, in that case the steady subsonic background solution is a constant state. Both isentropic and isothermal polytropic gases are considered.