Multidimensional transonic shock waves and free boundary problems
Multidimensional transonic shock waves and free boundary problems
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DOI：
10.1142/s166436072230002x
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发表时间：
2021-09
影响因子：
1.2
通讯作者：
Gui-Qiang G. Chen;M. Feldman
中科院分区：
文献类型：
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作者：
Gui-Qiang G. Chen;M. Feldman
本文讨论了可压缩流体力学中欧拉方程在分析多维跨音速激波时所引起的自由边界问题。本文综述了在多维跨音速激波分析和相应的可压缩欧拉方程及相关的混合型非线性偏微分方程的自由边界问题方面的一些最新进展。本文分析的非线性偏微分方程包括定常势流欧拉方程、定常全欧拉方程、定常势流欧拉方程以及相关的混合椭圆-双曲型非线性偏微分方程。跨声速激波问题包括稳定跨声速流过固体楔块的问题、激波反射-衍射的冯·诺依曼问题和非定常超音速流过固体楔块的prandtle - meyer问题。我们首先展示了如何将这些长期存在的多维跨音速激波问题表述为可压缩欧拉方程和相关的混合型非线性偏微分方程的自由边界问题。然后给出了求解这些自由边界问题的一种有效的非线性方法及相关思想和技术。该方法、思想和技术将有助于分析其他长期存在的和新出现的非线性偏微分方程的自由边界问题。
We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic–hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection–diffraction, and the Prandtl–Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.