American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021066

Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments

 1 School of Mathematical Sciences, MOE-LSC, and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China 2 Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070, China

* Corresponding author

Dedicated to Professor Shuxing Chen on the occasion of his 80th birthday

Received  February 2021 Revised  March 2021 Published  April 2021

Fund Project: The research was supported by Natural Science Foundation of China under Grant Nos. 11631008 and 11971308

This paper studies the uniqueness of steady 1-D shock solutions in a finite flat nozzle via vanishing viscosity arguments. It is proved that, for both barotropic gases and non-isentropic gases, the steady viscous shock solutions converge under the $\mathcal{L}^{1}$ norm. Hence only one shock solution of the inviscid Euler system could be the limit as the viscosity coefficient goes to $0$, which shows the uniqueness of the steady 1-D shock solution in a finite flat nozzle. Moreover, the position of the shock front for the limit shock solution is also obtained.

Citation: Beixiang Fang, Qin Zhao. Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021066
References:

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References:
Steady normal shocks in a flat nozzle
The velocity functions $u^{\varepsilon}(x)$ for different viscosity $\varepsilon>0$ and their limit as $\varepsilon \to 0+$
The graph of the function $f(u)$
The auxilliary functions for $f(u)$
Auxilliary points for $f(u)$
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