On the Well-Posedness of Periodic Strong Transonic Shocks in Divergent Nozzles

Abstract: This paper studies the $C<sup>1$ existence and dynamical stability of temporal periodic solutions involving strong transonic shocks for the quasi--one--dimensional full compressible Euler equations in diverging nozzles. We reveal the physical structural dissipation mechanism of strong transonic shocks hidden in the Rankine--Hugoniot conditions. This allows us to estimate the resonance at the shock boundary caused by the left acoustic waves in the subsonic downstream region. With this dissipation, we complete the proof by developing a fraction--step linearized iterative method.