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

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 \begin{document}$ \mathcal{L}^{1} $\end{document} norm. Hence only one shock solution of the inviscid Euler system could be the limit as the viscosity coefficient goes to \begin{document}$ 0 $\end{document} , 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.