The existence, uniqueness, and asymptotic behavior of steady transonic flows past a curved wedge, involving transonic shocks, governed by the two-dimensional full Euler equations are established. The stability of both weak and strong transonic shocks under the perturbation of both the upstream supersonic flow and the wedge boundary is proved. The problem is formulated as a one-phase free boundary problem, in which the transonic shock is treated as a free boundary. The full Euler equations are decomposed into two algebraic equations and a first-order elliptic system of two equations in Lagrangian coordinates. With careful elliptic estimates by using appropriate weighted H\"older norms, the iteration map is defined and analyzed, and the existence of its fixed point is established by performing the Schauder fixed point argument. The careful analysis of the asymptotic behavior of the solutions reveals particular characters of the full Euler equations.

中文翻译：

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全欧拉方程的跨音速流的稳定性和渐近行为

建立了稳定跨音速流的存在性、唯一性和渐近行为，通过弯曲楔形，涉及跨音速激波，由二维全欧拉方程控制。证明了在上游超音速流动和楔形边界扰动下弱和强跨音速激波的稳定性。该问题被表述为单相自由边界问题，其中跨音速激波被视为自由边界。完整的欧拉方程被分解为两个代数方程和一个在拉格朗日坐标中的两个方程的一阶椭圆系统。通过使用适当的加权 H 旧范数进行仔细的椭圆估计，定义和分析迭代图，并通过执行 Schauder 不动点论证建立其不动点的存在性。