This paper concerns the structural stability of smooth cylindrically symmetric transonic flows in a concentric cylinder. Both cylindrical and axi-symmetric perturbations are considered. The governing system here is of mixed elliptic–hyperbolic and changes type and the suitable formulation of boundary conditions at the boundaries is of great importance. First, we establish the existence and uniqueness of smooth cylindrical transonic spiral solutions with nonzero angular velocity and vorticity which are close to the background transonic flow with small perturbations of the Bernoulli’s function and the entropy at the outer cylinder and the flow angles at both the inner and outer cylinders independent of the symmetric axis, and it is shown that in this case, the sonic points of the flow are nonexceptional and noncharacteristically degenerate, and form a cylindrical surface. Second, we also prove the existence and uniqueness of axi-symmetric smooth transonic rotational flows which are adjacent to the background transonic flow, whose sonic points form an axi-symmetric surface. The key elements in our analysis are to utilize the deformation-curl decomposition for the steady Euler system to deal with the hyperbolicity in subsonic regions and to find an appropriate multiplier for the linearized second-order mixed type equations which are crucial to identify the suitable boundary conditions and to yield the important basic energy estimates.

中文翻译：

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关于非零角速度和涡度的一些光滑对称跨音速流

本文研究了同心圆柱体中光滑圆柱对称跨音速流动的结构稳定性。圆柱和轴对称扰动都被考虑在内。这里的控制系统是混合椭圆-双曲线和变化类型，边界条件的合适公式是非常重要的。首先，我们建立了具有非零角速度和涡量的光滑圆柱跨音速螺旋解的存在性和唯一性，它们接近背景跨音速流，并且伯努利函数和外圆柱的熵以及内圆柱的流动角都有小扰动。和独立于对称轴的外圆柱，并且表明在这种情况下，流动的声波点是非异常且非特征退化的，并形成一个圆柱面。其次，我们还证明了与背景跨音速流相邻的轴对称平滑跨音速旋转流的存在性和唯一性，其声波点形成轴对称表面。我们分析的关键要素是利用稳态欧拉系​​统的变形-卷曲分解来处理亚音速区域的双曲线，并为线性化二阶混合型方程找到合适的乘数，这对于确定合适的边界至关重要条件并产生重要的基本能量估计。