Prandtl-Meyer Reflection Configurations, Transonic Shocks, and Free Boundary Problems
Prandtl-Meyer Reflection Configurations, Transonic Shocks, and Free Boundary Problems
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发表时间：
2019-01
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通讯作者：
Myoungjean Bae;Gui-Qiang G. Chen;M. Feldman
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作者：
Myoungjean Bae;Gui-Qiang G. Chen;M. Feldman
我们关注撞击对称固体楔子的超音速流非定常全局解的普朗特-迈耶反射构型。 Prandtl (1936)首先利用激波极性分析表明，当稳定的超音速流撞击角度小于脱离角的楔块时，存在两种可能的稳态构型：稳态弱/强激波解，并推测稳态弱激波解是物理上允许的。在过去的八年里，关于稳定的弱/强冲击中的一种或两种在物理上是否可接受的基本问题一直存在激烈争论。另一方面，普朗特-迈耶反射构型是二维黎曼问题全局熵解结构中的核心构型，而黎曼解本身是局部构件，决定了一般熵解的局部结构、全局吸引子和大时间渐近态。从这个意义上说，我们必须了解反射构型，才能充分理解二维双曲守恒定律系统的全局熵解，包括熵解的可接受性问题。在本专着中，我们解决了这个长期悬而未决的问题，并提出了我们的分析，以建立稳态弱激波解的稳定性定理，作为非稳态势流的普朗特-迈耶反射配置的长期渐近性，适用于所有物理参数直至脱离角。为了实现这些，我们首先将问题重新表述为涉及跨音速激波的自由边界问题，然后获得适当的单调性属性和可接受解的统一先验估计，这使我们能够采用 Leray-Schauder 度论证来完成直到脱离角的所有物理参数的理论。
We are concerned with the Prandtl-Meyer reflection configurations of unsteady global solutions for supersonic flow impinging upon a symmetric solid wedge. Prandtl (1936) first employed the shock polar analysis to show that there are two possible steady configurations: the steady weak/strong shock solutions, when a steady supersonic flow impinges upon the wedge whose angle is less than the detachment angle, and then conjectured that the steady weak shock solution is physically admissible. The fundamental issue of whether one or both of the steady wea/strong shocks are physically admissible has been vigorously debated over the past eight decades. On the other hand, the Prandtl-Meyer reflection configurations are core configurations in the structure of global entropy solutions of the 2-D Riemann problem, while the Riemann solutions themselves are local building blocks and determine local structures, global attractors, and large-time asymptotic states of general entropy solutions. In this sense, we have to understand the reflection configurations in order to understand fully the global entropy solutions of 2-D hyperbolic systems of conservation laws, including the admissibility issue for the entropy solutions. In this monograph, we address this longstanding open issue and present our analysis to establish the stability theorem for the steady weak shock solutions as the long-time asymptotics of the Prandtl-Meyer reflection configurations for unsteady potential flow for all the physical parameters up to the detachment angle. To achieve these, we first reformulate the problem as a free boundary problem involving transonic shocks and then obtain appropriate monotonicity properties and uniform a priori estimates for admissible solutions, which allow us to employ the Leray-Schauder degree argument to complete the theory for all the physical parameters up to the detachment angle.