Global Steady Subsonic Flows through Infinitely Long Nozzles for the Full Euler Equations

TL;DR

This paper proves the existence and uniqueness of global steady subsonic flows through infinitely long nozzles for the full Euler equations, under small entropy and Bernoulli oscillations, clarifying flow behavior and critical flux.

Contribution

It establishes the first rigorous existence and uniqueness results for subsonic Euler flows in general nozzles with small upstream oscillations.

Findings

Existence of unique global subsonic solutions under small oscillations.

Characterization of asymptotic behavior at upstream and downstream.

Identification of the critical mass flux preventing supersonic bubbles.

Abstract

We are concerned with global steady subsonic flows through general infinitely long nozzles for the full Euler equations. The problem is formulated as a boundary value problem in the unbounded domain for a nonlinear elliptic equation of second order in terms of the stream function. It is established that, when the oscillation of the entropy and Bernoulli functions at the upstream is sufficiently small in $C^{1,1}$ and the mass flux is in a suitable regime, there exists a unique global subsonic solution in a suitable class of general nozzles. The assumptions are required to prevent from the occurrence of supersonic bubbles inside the nozzles. The asymptotic behavior of subsonic flows at the downstream and upstream, as well as the critical mass flux, have been clarified.