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arXiv · 2407.09917

Transonic shock solutions for steady 3-D axisymmetric full Euler flows with large swirl velocity in a finite cylindrical nozzle

Beixiang Fang·Xin Gao·Wei Xiang·Qin Zhao
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Abstract

This paper concerns the existence and location of three-dimensional axisymmetric transonic shocks with large swirl velocity for shock solutions of the steady compressible full Euler system in a cylindrical nozzle with prescribed receiver pressure. As far as we know, it is the first mathematical result on the three-dimensional transonic shock with either large vorticity or large swirl velocity. One of the key difficulties is the fact that the Euler system is elliptic-hyperbolic composite for the flow behind the shock front, and its elliptic part and hyperbolic part are strongly coupled in the lower order terms because of the large swirl velocity, such that they cannot be simply decoupled in the principal parts as the case for the flow without swirls or with small swirl velocity. New decomposition techniques for the elliptic-hyperbolic composite system are developed to deal with this difficulty, and the solvability condition for the boundary value problem is deduced to determine the location of the shock front. It finally turns out that the non-zero swirl velocity, which brings new challenging difficulties in the analysis, plays an essential and fundamental role in determining the location of the shock front. Another key difficulty in the analysis is that there are no readily established shock solutions available. Non-trivial special shock solutions are first constructed as background solutions under the assumption that the flow parameters depend only on the radial distance to the symmetric axis. Necessary and sufficient conditions for the existence of such solutions are also found. Even though the shock location can be arbitrarily shifted for the special shock solutions, it will be shown that the shock solution with a determined shock position can be established as the boundary data are perturbations of one of the established special shock solutions under certain conditions.

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Beixiang Fang, Xin Gao, Wei Xiang, Qin Zhao. 2026-01-21. Transonic shock solutions for steady 3-D axisymmetric full Euler flows with large swirl velocity in a finite cylindrical nozzle. https://arxiv.org/abs/2407.09917

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