Convexity of Self-Similar Transonic Shocks and Free Boundaries for the Euler Equations for Potential Flow
Convexity of Self-Similar Transonic Shocks and Free Boundaries for the Euler Equations for Potential Flow
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DOI：
10.1007/s00205-020-01528-0
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发表时间：
2020-06
影响因子：
2.5
通讯作者：
Gui-Qiang G. Chen;M. Feldman;Wei Xiang
中科院分区：
文献类型：
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作者：
Gui-Qiang G. Chen;M. Feldman;Wei Xiang
我们关注跨音速激波作为可压缩流体流动二维自相似坐标中的自由边界的几何性质，这不仅对于理解连续介质力学中流体运动的几何结构和稳定性很重要，而且也是多维守恒定律数学理论的基础。自相似势流欧拉方程的跨音速激波将所考虑的域中相应非线性偏微分方程的自相似解的椭圆（亚音速）和双曲（超音速）相分开，其中跨音速激波的位置是先验未知的。我们首先建立了一个通用框架，在该框架下，自相似的跨音速激波作为自由边界被证明是一致凸的，然后应用该框架来证明跨音速激波在两个长期存在的基本激波问题中的一致凸性：激波反射-楔子衍射和超音速流通过固体坡道的普朗特-迈耶反射。为了实现这一目标，我们的方法是利用解的潜在非局部属性和势流方程的自由边界。
We are concerned with geometric properties of transonic shocks as free boundaries in two-dimensional self-similar coordinates for compressible fluid flows, which are not only important for the understanding of geometric structure and stability of fluid motions in continuum mechanics, but are also fundamental in the mathematical theory of multidimensional conservation laws. A transonic shock for the Euler equations for self-similar potential flow separates elliptic (subsonic) and hyperbolic (supersonic) phases of the self-similar solution of the corresponding nonlinear partial differential equation in a domain under consideration, in which the location of the transonic shock isaprioriunknown. We first develop a general framework under which self-similar transonic shocks, as free boundaries, are proved to be uniformly convex, and then apply this framework to prove the uniform convexity of transonic shocks in the two longstanding fundamental shock problems—the shock reflection–diffraction by wedges and the Prandtl–Meyer reflection for supersonic flows past solid ramps. To achieve this, our approach is to exploit underlying nonlocal properties of the solution and the free boundary for the potential flow equation.