Validity of the supersonic-inflow existence theorem without the Mach-number assumptions

Determine whether the existence conclusions of Theorem 1.2 remain valid for the one-dimensional steady Euler system with heat conductivity in a finite nozzle when both sufficient conditions (S1) and (S2) on the specific-heat ratio and incoming Mach number fail.

Background

Theorem 1.2 establishes existence of purely supersonic or transonic heat-conductive solutions for prescribed uniform supersonic inflow and exit pressure, but its proof imposes either condition (S1) or condition (S2) to avoid degeneracy in the principal coefficient of the reformulated equations for certain exit-pressure ranges.

The paper notes that these assumptions are sufficient rather than universally necessary in some parameter regimes. It leaves unresolved whether the theorem's existence conclusions continue to hold in the remaining parameter regime where neither (S1) nor (S2) is satisfied.