Three Dimensional Steady Subsonic Euler Flows in Bounded Nozzles

TL;DR

This paper investigates the existence and uniqueness of three-dimensional steady subsonic Euler flows in bounded nozzles, establishing conditions for potential flow and analyzing flow behavior near critical momentum values.

Contribution

It introduces a new formulation for the Euler system and provides rigorous analysis for the existence, uniqueness, and convergence of subsonic flows in nozzles under various conditions.

Findings

Existence of unique subsonic potential flow under specific conditions.

Flows converge to subsonic-sonic flow as momentum approaches a critical value.

Small vorticity and Bernoulli variation ensure existence of subsonic Euler flows.

Abstract

In this paper, we study the existence and uniqueness of three dimensional steady Euler flows in rectangular nozzles when prescribing normal component of momentum at both the entrance and exit. If, in addition, the normal component of the voriticity and the variation of Bernoulli's function at the exit are both zero, then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number. As the magnitude of the normal component of the momentum approaches the critical number, the associated flows converge to a subsonic-sonic flow. Furthermore, when the normal component of vorticity and the variation of Bernoulli's function are both small, the existence of subsonic Euler flows is established. The proof of these results is based on a new formulation for the Euler system, a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions, detailed study for a linear div-curl system, and delicate estimate for the transport equations.