SIAM Journal on Mathematical Analysis, Volume 53, Issue 1, Page 279-308, January 2021.
In this paper, we prove the structural stability of the transonic shocks for three-dimensional axisymmetric Euler system with swirl velocity under the perturbations for the incoming supersonic flow, the nozzle boundary, and the exit pressure. Compared with the known results on the stability of transonic shocks, one of the major difficulties for the axisymmetric flows with swirls is that corner singularities near the intersection point of the shock surface and nozzle boundary and the artificial singularity near the axis appear simultaneously. One of the key points in the analysis for this paper is the introduction of an invertible modified Lagrangian transformation which can straighten the streamlines in the whole nozzle and help to represent the solutions of transport equations explicitly. Furthermore, the simple but useful modified Lagrangian transformation makes the treatment for the singularities near the axis easy and clean. Such a technique may be helpful in studies of other problems for axisymmetric flows.

中文翻译：

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发散的三维轴对称扰动喷嘴中跨音速冲击问题的结构稳定性

SIAM数学分析杂志，第53卷，第1期，第279-308页，2021年1月。
在本文中，我们证明了在输入超音速流，喷嘴边界和出口压力的扰动下，具有涡旋速度的三维轴对称Euler系统的跨音速激波的结构稳定性。与关于跨音速冲击稳定性的已知结果相比，具有漩涡的轴对称流的主要困难之一是，在冲击表面与喷嘴边界的交点附近的角奇点和在轴附近的人为奇点同时出现。本文分析的关键点之一是引入了可逆的修正拉格朗日变换，该变换可以拉直整个喷嘴中的流线，并有助于明确表示输运方程的解。此外，简单但有用的修改后的拉格朗日变换使对轴附近的奇异点的处理变得容易且干净。这种技术可能有助于研究轴对称流动的其他问题。