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Volume 41, Issue 3
Dynamical Stability of Transonic Shock Solutions to Non-Isentropic Euler Equations

Ben Duan & Yan Zhou

DOI: 10.4208/cmr.2025-0032

Commun. Math. Res., 41 (2025), pp. 250-270.

Published online: 2025-09

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  • Abstract

In this paper, we investigate the dynamical stability of transonic shock solutions for the full compressible Euler system in a two dimensional nozzle with a symmetric divergent part. Building upon the existence and uniqueness results for steady symmetric transonic shock solutions to the non-isentropic Euler system established in [Z.P. Xin and H.C. Yin, The transonic shock in a nozzle, 2-D and 3-D complete Euler systems, J. Differential Equations 245 (2008)], we prove the dynamical stability of the transonic shock solutions under small perturbations. More precisely, if the initial unsteady transonic flow is located in the symmetric divergent part of the nozzle and the flow is a symmetric small perturbation of the steady transonic flow, we use the characteristic method to establish the dynamical stability.

  • Keywords

Euler equation, transonic shock, dynamical stability.

  • AMS Subject Headings

35B35, 35L65, 76H05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-41-250, author = {Duan , Ben and Zhou , Yan}, title = {Dynamical Stability of Transonic Shock Solutions to Non-Isentropic Euler Equations}, journal = {Communications in Mathematical Research }, year = {2025}, volume = {41}, number = {3}, pages = {250--270}, abstract = {

In this paper, we investigate the dynamical stability of transonic shock solutions for the full compressible Euler system in a two dimensional nozzle with a symmetric divergent part. Building upon the existence and uniqueness results for steady symmetric transonic shock solutions to the non-isentropic Euler system established in [Z.P. Xin and H.C. Yin, The transonic shock in a nozzle, 2-D and 3-D complete Euler systems, J. Differential Equations 245 (2008)], we prove the dynamical stability of the transonic shock solutions under small perturbations. More precisely, if the initial unsteady transonic flow is located in the symmetric divergent part of the nozzle and the flow is a symmetric small perturbation of the steady transonic flow, we use the characteristic method to establish the dynamical stability.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2025-0032}, url = {http://global-sci.org/intro/article_detail/cmr/24469.html} }
TY - JOUR T1 - Dynamical Stability of Transonic Shock Solutions to Non-Isentropic Euler Equations AU - Duan , Ben AU - Zhou , Yan JO - Communications in Mathematical Research VL - 3 SP - 250 EP - 270 PY - 2025 DA - 2025/09 SN - 41 DO - http://doi.org/10.4208/cmr.2025-0032 UR - https://global-sci.org/intro/article_detail/cmr/24469.html KW - Euler equation, transonic shock, dynamical stability. AB -

In this paper, we investigate the dynamical stability of transonic shock solutions for the full compressible Euler system in a two dimensional nozzle with a symmetric divergent part. Building upon the existence and uniqueness results for steady symmetric transonic shock solutions to the non-isentropic Euler system established in [Z.P. Xin and H.C. Yin, The transonic shock in a nozzle, 2-D and 3-D complete Euler systems, J. Differential Equations 245 (2008)], we prove the dynamical stability of the transonic shock solutions under small perturbations. More precisely, if the initial unsteady transonic flow is located in the symmetric divergent part of the nozzle and the flow is a symmetric small perturbation of the steady transonic flow, we use the characteristic method to establish the dynamical stability.

Duan , Ben and Zhou , Yan. (2025). Dynamical Stability of Transonic Shock Solutions to Non-Isentropic Euler Equations. Communications in Mathematical Research . 41 (3). 250-270. doi:10.4208/cmr.2025-0032
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