Structural Stability of 3D Axisymmetric Steady Subsonic Euler Flows in Finitely Long Nozzles with Variable End Pressures

Abstract

As a continuation of [Li, J. and Wang, Y. N., Structural stability of steady subsonic Euler flows in 2D finitely long nozzles with variable end pressure, J. Differential Equations, 413, 2014, 70–109], in this paper, the authors study the structural stability of three dimensional axisymmetric steady subsonic Euler flows in finitely long curved nozzles. The reference flow is a general subsonic shear flow in a three dimensional regular cylindrical nozzle with general size of vorticity and without stagnation points. The problem is described by the well-known steady compressible Euler system. With a class of admissible physical conditions and prescribed pressure at the entrance and the exit of the nozzle respectively, they establish the structural stability of this kind of axisymmetric subsonic shear flow with no-zero swirl velocity. Due to the hyperbolic-elliptic coupled form of the Euler system in subsonic regions, the problem is reformulated via a twofold normalized process, including straightening the lateral boundary of the nozzle under the natural Cartesian coordinates and reformulating the problem under the cylindrical coordinates. Accordingly, the Euler system is decoupled into an elliptic mode and three hyperbolic modes with some artificial singular terms under the cylindrical coordinates. The elliptic mode is a mixed type boundary value problem of first order elliptic system for the pressure and the radial velocity angle. Meanwhile, the hyperbolic modes are transport type to control the total energy, the specific entropy and the swirl velocity, respectively. The estimates as well as well-posedness are executed in a Banach space with optimal regularity under the natural Cartesian coordinates in place of the cylindrical coordinates. The authors develop a systematic framework to deal with the artificial singularity and the non-zero swirl velocity in three dimensional axisymmetric case. Their strategy is helpful for other three dimensional problems under axisymmetry.

Keywords

Steady compressible Euler system / Subsonic shear flow / First order elliptic system / Structural stability / Axisymmetry / 35B35 / 35J56 / 35M12 / 76G25

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