“…Phenomena involving transonic flows and transonic flows with shocks are fundamental to fluid dynamics, especially gas dynamics, and have been studied extensively in the literature [4,10,11,13,23,24,25,27,28,29,30,31] and the references therein). Profound understanding has been achieved both physically and mathematically by Morawetz [27,28,30] and others [4,10] on smooth transonic flows. Most previous studies of transonic flows with shocks involve either experimental or numerical simulations or analysis of special wave patterns [4,10,12,16], except the rigorous results on the existence and stability of the quasi one-dimensional transonic shocks; see [11,23].…”