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Dynamic stability of transonic shock solutions for one-dimensional Euler equations with external forcing

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Abstract

In this paper, we are concerned with the global in time dynamic stability of steady transonic shock passing a flat nozzle with external forcing. It is shown that the steady transonic shock solution with external forcing is dynamically stable under the small perturbations of the initial data. Furthermore, the subsonic flow and shock profile exponentially decay to the corresponding steady state as time goes to infinity. The dynamic stability of the steady transonic shock for the Euler system is formulated as a nonlinear free boundary value problem with nonlinear boundary conditions. An important part of the proof is the establishment of exponentially decaying energy estimates for the associated linearized problem.

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Acknowledgements

The author is very grateful to professor Shangkun Weng for providing this question and for the very useful discussion.

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  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei Province, People’s Republic of China

    Wengang Yang

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  1. Wengang Yang
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Yang, W. Dynamic stability of transonic shock solutions for one-dimensional Euler equations with external forcing. J. Evol. Equ. 23, 16 (2023). https://doi.org/10.1007/s00028-023-00867-1

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  • DOI: https://doi.org/10.1007/s00028-023-00867-1

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