This paper concerns the existence of transonic shock solutions to the 2-D steady compressible Euler system in an almost flat finite nozzle ( in the sense that it is a generic small perturbation of a flat one ), under physical boundary conditions proposed by Courant-Friedrichs in \cite{CourantFriedrichs1948}, in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock-front is one of the most desirable information one would like to determine. However, the location of the normal shock-front in a flat nozzle can be anywhere in the nozzle so that it provides little information on the possible location of the shock-front when the nozzle's boundary is perturbed. So one of the key difficulties in looking for transonic shock solutions is to determine the shock-front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution will be taken as an initial approximation for the transonic shock solution. In this paper, a sufficient condition in terms of the geometry of the nozzle and the given exit pressure is derived which yields the existence of the solutions to the proposed free boundary problem. Once an initial approximation is obtained, a further nonlinear iteration could be constructed and proved to lead to a transonic shock solution.

中文翻译：

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具有规定接收压力的几乎扁平的有限喷嘴中稳态欧拉流的跨音速激波前沿的容许位置

本文涉及在 Courant-Friedrichs 提出的物理边界条件下，在几乎扁平的有限喷嘴（从某种意义上说，它是扁平喷嘴的一般小扰动）中二维稳定可压缩欧拉系统的跨音速激波解的存在在 \cite{CourantFriedrichs1948} 中，其中接收器压力在喷嘴出口处规定。在由此产生的自由边界问题中，激波前沿的位置是人们最想要确定的信息之一。然而，扁平喷嘴中正常激波前沿的位置可以位于喷嘴中的任何位置，因此当喷嘴边界受到扰动时，它几乎不能提供有关激波前沿可能位置的信息。因此，寻找跨音速激波解决方案的主要困难之一是确定激波前沿。为此，将提出线性化欧拉系统的自由边界问题，其解将作为跨音速激波解的初始近似。在本文中，根据喷嘴的几何形状和给定​​的出口压力导出了一个充分条件，它产生了所提出的自由边界问题的解的存在。一旦获得了初始近似值，就可以构建进一步的非线性迭代并证明可以得到跨音速激波解。就喷嘴的几何形状和给定​​的出口压力而言，导出了一个充分条件，这产生了所提出的自由边界问题的解的存在。一旦获得了初始近似值，就可以构建进一步的非线性迭代并证明可以得到跨音速激波解。就喷嘴的几何形状和给定​​的出口压力而言，导出了一个充分条件，这产生了所提出的自由边界问题的解的存在。一旦获得了初始近似值，就可以构建进一步的非线性迭代并证明可以得到跨音速激波解。