Validity of the subsonic-inflow existence theorem without the technical assumptions

Verify whether the existence conclusions of Theorem 1.9 remain valid for the one-dimensional steady Euler system with heat conductivity in a finite nozzle when neither sufficient condition (S1) nor (S2) holds.

Background

Theorem 1.9 proves existence of transonic or purely subsonic heat-conductive solutions when the prescribed inflow is the uniform subsonic state. The proof retains the sufficient conditions (S1) and (S2) introduced for the supersonic-inflow case.

The authors explicitly state that it is unresolved whether the theorem remains true without either condition, leaving the existence theory incomplete for subsonic incoming flow in that parameter regime.