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Stability of Attached Transonic Shocks in Steady Potential Flow past Three-Dimensional Wedges

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Abstract

We develop a new approach and employ it to establish the global existence and nonlinear structural stability of attached weak transonic shocks in steady potential flow past three-dimensional wedges; in particular, the restriction that the perturbations are away from the wedge edge in the previous results is removed. One of the key ingredients is to identify a good direction of the boundary operator of a boundary condition of the shock along the wedge edge, based on the non-obliqueness of the boundary condition for the weak shock on the edge. With the identification of this direction, an additional boundary condition on the wedge edge can be assigned to make sure that the shock is attached on the edge and linearly stable under small perturbations. Based on the linear stability, we introduce an iteration scheme and prove that there exists a unique fixed point of the iteration scheme, which leads to the global existence and nonlinear structural stability of the attached weak transonic shock. This approach is based on neither the hodograph transformation nor the spectrum analysis, and should be useful for other problems with similar difficulties.

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Acknowledgements

The research of Gui-Qiang G. Chen was supported in part by the UK Engineering and Physical Sciences Research Council Award EP/L015811/1, and the Royal Society–Wolfson Research Merit Award WM090014 (UK). Jun Chen’s research was supported in part by Yichun University Doctoral Start-up Grant 207-3360119008 and NSFC Regional Science Funds 12061080 (China). The research of Wei **ang was supported in part by the Research Grants Council of the HKSAR, China (Project CityU 11303518, Project CityU 11304820, and Project CityU 11300021).

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

    Gui-Qiang G. Chen

  2. Center of Applied Mathematics, Yichun University, Yichun, 336000, Jiangxi, China

    Jun Chen

  3. Department of Mathematics, City University of Hong Kong, Hong Kong, China

    Wei **ang

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  1. Gui-Qiang G. Chen
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  2. Jun Chen
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Correspondence to Gui-Qiang G. Chen.

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Chen, GQ.G., Chen, J. & **ang, W. Stability of Attached Transonic Shocks in Steady Potential Flow past Three-Dimensional Wedges. Commun. Math. Phys. 387, 111–138 (2021). https://doi.org/10.1007/s00220-021-04168-x

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