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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2404.14652 |
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| _version_ | 1866918519598219264 |
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| author | Zhang, Zihao |
| author_facet | Zhang, Zihao |
| contents | We establish the existence and stability of the transonic shock solution to three-dimensional axisymmetric Euler system with an external force in a cylinder under perturbations of the incoming supersonic flow, the exit pressure, the external force and the nozzle wall. The external force has a stabilization effect on the transonic shock in the straight cylinder and the shock position is uniquely determined. The main difficulties for the axisymmetric flows are the corner singularities near the intersection point of the shock surface and the nozzle wall and the singularity near the symmetry axis. An invertible modified Lagrangian transformation is introduced to overcome these difficulties and straighten the streamline. One of the key elements in the analysis is to decompose the hyperbolic and elliptic modes for the steady axisymmetric Euler system with an external force in terms of the deformation and vorticity. Another one is an equivalent reformulation of the Rankine-Hugoniot conditions so that the shock front is uniquely determined by an algebraic equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14652 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transonic shocks for three-dimensional axisymmetric Euler flows with an external force in a perturbed cylinder Zhang, Zihao Analysis of PDEs We establish the existence and stability of the transonic shock solution to three-dimensional axisymmetric Euler system with an external force in a cylinder under perturbations of the incoming supersonic flow, the exit pressure, the external force and the nozzle wall. The external force has a stabilization effect on the transonic shock in the straight cylinder and the shock position is uniquely determined. The main difficulties for the axisymmetric flows are the corner singularities near the intersection point of the shock surface and the nozzle wall and the singularity near the symmetry axis. An invertible modified Lagrangian transformation is introduced to overcome these difficulties and straighten the streamline. One of the key elements in the analysis is to decompose the hyperbolic and elliptic modes for the steady axisymmetric Euler system with an external force in terms of the deformation and vorticity. Another one is an equivalent reformulation of the Rankine-Hugoniot conditions so that the shock front is uniquely determined by an algebraic equation. |
| title | Transonic shocks for three-dimensional axisymmetric Euler flows with an external force in a perturbed cylinder |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.14652 |