This paper is concerned with the well-posedness theory of the impact of a subsonic axially symmetric jet emerging from a semi-infinitely long nozzle, onto a rigid wall. The fluid motion is described by the steady isentropic Euler system. We showed that there exists a critical value $M_{cr} > 0$, if the given mass flux is less than $M_{cr}$, there exists a unique smooth subsonic axially symmetric jet issuing from the given semi-infinitely long nozzle and hitting a given uneven wall. The surface of the axially symmetric impinging jet is a free boundary, which detaches from the edge of the nozzle smoothly. It is showed that a unique suitable choice of the pressure difference between the chamber and the atmosphere guarantees the continuous fit condition of the free boundary. Moreover, the asymptotic behaviors and the decay properties of the impinging jet and the free surface in downstream were also obtained. The main results in this paper solved the open problem on the well-posedness of the compressible axially symmetric impinging jet, which has proposed by A. Friedman in Chapter 16 in [26]. The key ingredient of our proof is based on the variational method to the quasilinear elliptic equation with the Bernoulli’s type free boundaries.

中文翻译：

---

轴向对称可压缩亚音速射流撞击无限壁的存在与唯一性

本文关注的是从半无限长的喷嘴喷出的亚音速轴向对称射流对刚性壁的冲击的适定性理论。稳态等熵欧拉系统描述了流体运动。我们表明存在一个临界值$ M_ {cr}> 0 $，如果给定的质量通量小于$ M_ {cr} $，则存在一个从给定的半无限长喷嘴发出的独特的平滑亚音速轴向对称射流撞到给定的不平坦的墙。轴向对称的冲击射流的表面是自由边界，该边界平滑地从喷嘴的边缘分离。结果表明，腔室与大气之间的压差的独特合适选择保证了自由边界的连续拟合条件。而且，还获得了冲击射流和下游自由表面的渐近行为和衰减特性。本文的主要结果解决了A. Friedman在[26]的第16章中提出的关于可压缩轴向对称冲击射流的适定性的开放性问题。我们证明的关键要素是基于具有伯努利类型自由边界的拟线性椭圆方程的变分方法。