This paper concerns the existence and uniqueness of a two-dimensional compressible subsonic cavity flow behind a given obstacle with a free surface in an infinitely long nozzle. More precisely, for any given atmospheric pressure p_e > 0, we first show that there exists a critical value m_cr for the incoming mass flux m₀ such that if m₀ < m_cr, then there exists a unique smooth subsonic cavity flow behind the given obstacle, and the free surface initiates smoothly from the vertex of the obstacle and goes to infinity at the far field. Second, we establish some fundamental properties of the cavity flow, such as the positivity of the horizontal velocity and the asymptotic behavior of the flow at the far field. Finally, we obtain that the vertical velocity is positive under the corresponding assumptions on the upper nozzle wall and the upper surface of the obstacle.

中文翻译：

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喷嘴中可压缩的亚音速腔流

本文涉及在无限长喷嘴内具有给定障碍物且具有自由表面的二维可压缩亚音速腔流的存在和唯一性。更准确地说，对于任何给定的大气压p _e > 0，我们首先表明对于进入的质量通量m ₀存在一个临界值m _cr，使得如果m ₀ < m _cr，然后在给定的障碍物后面存在一个独特的亚音速空洞，并且自由表面从障碍物的顶点平滑地开始，并在远场处达到无穷大。其次，我们建立了腔流的一些基本属性，例如水平速度的正性和远场流的渐近行为。最终，在相应的假设下，在上喷嘴壁和障碍物的上表面上，垂直速度为正。