This paper concerns with the existence of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle, which are governed by the Euler equations with the slip boundary condition on the wall of the nozzle and a receiver pressure at the exit. Mathematically, it can be formulated as a free boundary problem with the shock front being the free boundary to be determined. In dealing with the free boundary problem, one of the key points is determining the position of the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution gives an initial approximating position of the shock front. Compared with the 2-D case, new difficulties arise due to the additional 0-order terms and singularities along the symmetric axis. New observations and careful analysis will be done to overcome these difficulties. Once the initial approximation is obtained, a nonlinear iteration scheme can be carried out, which converges to a transonic shock solution to the problem.

本文关注的是在3-D轴对称圆柱喷嘴中存在稳定的Euler流动的跨音速冲击，这由Euler方程控制，喷嘴壁上有滑移边界条件，出口处有接收器压力。从数学上讲，它可以表述为自由边界问题，其中激波锋为待确定的自由边界。在处理自由边界问题时，关键点之一是确定激波锋的位置。为此，将提出线性化Euler系统的自由边界问题，其解给出了激波锋的初始近似位置。与二维情况相比，由于沿对称轴的附加0阶项和奇异性而产生了新的困难。将进行新的观察和仔细的分析以克服这些困难。一旦获得初始近似值，就可以执行非线性迭代方案，该方案可以收敛到该问题的跨音速激波解。