Global existence and stability of temporal periodic solution to non-isentropic compressible Euler equations with a source term

Shuyue Ma; Jiawei Sun; Huimin Yu; Shuyue Ma; Jiawei Sun; Huimin Yu

doi:10.3934/cam.2023013

Communications in Analysis and Mechanics

2023, Volume 15, Issue 2: 245-266. doi: 10.3934/cam.2023013

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Research article

Global existence and stability of temporal periodic solution to non-isentropic compressible Euler equations with a source term

School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China

Received: 23 April 2023 Revised: 16 May 2023 Accepted: 25 May 2023 Published: 08 June 2023
35Q31, 35B10, 35A01

In this paper, the 1-D compressible non-isentropic Euler equations with the source term $ \beta\rho|u|^ \alpha u $ in a bounded domain are considered. First, we study the existence of steady flows which can keep the upstream supersonic or subsonic state. Then, by wave decomposition and uniform prior estimations, we prove the global existence and stability of smooth solutions under small perturbations around the steady supersonic flow. Moreover, we get that the smooth supersonic solution is a temporal periodic solution with the same period as the boundary, after a certain start-up time, once the boundary conditions are temporal periodic.
- compressible Euler equations,
- friction,
- temporal periodic solutions,
- supersonic flow,
- global existence,
- stability
Citation: Shuyue Ma, Jiawei Sun, Huimin Yu. Global existence and stability of temporal periodic solution to non-isentropic compressible Euler equations with a source term[J]. Communications in Analysis and Mechanics, 2023, 15(2): 245-266. doi: 10.3934/cam.2023013

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Abstract

In this paper, the 1-D compressible non-isentropic Euler equations with the source term $ \beta\rho|u|^ \alpha u $ in a bounded domain are considered. First, we study the existence of steady flows which can keep the upstream supersonic or subsonic state. Then, by wave decomposition and uniform prior estimations, we prove the global existence and stability of smooth solutions under small perturbations around the steady supersonic flow. Moreover, we get that the smooth supersonic solution is a temporal periodic solution with the same period as the boundary, after a certain start-up time, once the boundary conditions are temporal periodic.

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