非线性波系中凸棱楔对激波的绕射
Shock Diffraction by Convex Cornered Wedges for the Nonlinear Wave System
摘要：
本文研究了在非线性波系支配下的可压缩流体中二维凸形角楔对激波绕射的严格数学分析。这个激波绕射问题可以表示为无界区域上的混合椭圆-双曲型二阶非线性偏微分方程的边值问题。它可以进一步转化为二阶非线性退化椭圆型方程的自由边界问题。我们建立了这个激波绕射问题的第一个整体存在性和正则性理论。特别是，我们建立的解决方案的最佳规律是$C^{0,1}$整个退化声波边界。为了实现这一点，我们开发了几个数学思想和技术，这也是有用的其他相关问题，涉及类似的分析困难。
英文摘要：
We are concerned with rigorous mathematical analysis of shock diffraction by two-dimensional convex cornered wedges in compressible fluid flow governed by the nonlinear wave system. This shock diffraction problem can be formulated as a boundary value problem for second-order nonlinear partial differential equations of mixed elliptic-hyperbolic type in an unbounded domain. It can be further reformulated as a free boundary problem for nonlinear degenerate elliptic equations of second order. We establish a first global theory of existence and regularity for this shock diffraction problem. In particular, we establish that the optimal regularity for the solution is $C^{0,1}$ across the degenerate sonic boundary. To achieve this, we develop several mathematical ideas and techniques, which are also useful for other related problems involving similar analytical difficulties.
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