Stability of Stationary Subsonic Compressible Euler Flows with Mass-Additions in Two-Dimensional Straight Ducts

TL;DR

This paper proves the existence, uniqueness, and stability of stationary subsonic compressible Euler flows with mass-additions in 2D ducts, introducing a new decomposition and iteration scheme to handle complex mode interactions.

Contribution

It develops a novel decomposition and nonlinear iteration method to analyze the stability of Euler flows with mass-additions, overcoming challenges posed by nonlocal terms and mode interactions.

Findings

Established existence and stability of solutions.

Developed a new decomposition and iteration scheme.

Analyzed complex interactions between elliptic and hyperbolic modes.

Abstract

We show existence, uniqueness and stability for a family of stationary subsonic compressible Euler flows with mass-additions in two-dimensional rectilinear ducts, subjected to suitable time-independent multi-dimensional boundary conditions at the entrances and exits.The stationary subsonic Euler equations consist a quasi-linear system of elliptic-hyperbolic composite-mixed type, while addition-of-mass destructs the usual methods based upon conservation of mass and Lagrangian coordinates to separate the elliptical and hyperbolic modes of the system. We establish a new decomposition and nonlinear iteration scheme to overcome this major difficulty. It reveals that mass-additions introduce very strong interactions in the elliptic and hyperbolic modes, and lead to a class of second-order elliptic equations with multiple integral nonlocal terms. The linearized problem is solved by studying algebraicand analytical properties of infinite weakly coupled boundary-value problems of ordinary differential equations, each with multiple nonlocal terms, after applications of Fourier analysis methods.