Structural stability of steady subsonic Rayleigh flow in a two-dimensional finite length straight nozzle

This paper investigates the steady gas flow in a finite-length straight nozzle governed by two-dimensional non-isentropic compressible Euler equations with heat transfer effects, known in engineering as Rayleigh flow. We formulate a boundary value problem derived from transonic shocks and construct a class of non-trivial subsonic flow special solutions in a constant cross-section straight nozzle. By proving the structural stability of this special solution under two-dimensional perturbations of upstream flow and downstream back pressures satisfying certain symmetry conditions, the rationality of the proposed boundary value problem is demonstrated. For subsonic flows, the two-dimensional steady compressible Euler equations represent typical quasi-linear elliptic-hyperbolic complex systems. We present a decomposition lemma for the mathematical model, which effectively separates the elliptic and hyperbolic modes in the steady subsonic Euler equations. It is shown that heat transfer effects introduce significant interactions between elliptic and hyperbolic modes, leading to a class of second-order elliptic equations with multiple integral-type non-local terms. The solution to the linearized problem is obtained by analyzing the algebraic and analytical properties of infinitely weakly coupled boundary value problems for integro-differential equations via Fourier analysis methods.