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Transonic Evaporation Waves in a Spherically Symmetric Nozzle

This paper studies the liquid-to-vapor phase transition in a cone-shaped nozzle. Using the geometric method presented in [P. Szmolyan and M. Wechselberger, J. Differential Equations, 200 (2004), pp. 69--104], [M. Wechselberger and G. Pettet, Nonlinearity, 23 (2010), pp. 1949--1969], we further develop results on subsonic and supersonic evaporation waves in [H. Fan and X.-B. Lin, SIAM J. Math. Anal., 44 (2012), pp. 405--436] to transonic waves. It is known that transonic waves do not exist if restricted solely to the slow system on the slow manifolds [H. Fan and X.-B. Lin, SIAM J. Math. Anal., 44 (2012), pp. 405--436]. Thus we consider the existence of transonic waves that include layer solutions of the fast system that cross or connect to the sonic surface. In particular, we are able to show the existence and uniqueness of evaporation waves that cross from supersonic to subsonic regions and evaporation waves that connect from the subsonic region to the sonic surface and then continue onto the supersonic b...